Evaluate The Expression \( \frac{\cos A (1 + \cos A)}{\cos B (1 - \sin B)} \) If \( A + B = 90^\circ \) And \( \sin A = \frac{11}{61} \).

In this article, we delve into a fascinating trigonometric problem involving complementary angles and the evaluation of a complex expression. Specifically, we are given that ${ \hat{A} + \hat{B} = 90^\circ }$ and ${ \sin A = \frac{11}{61} }$. Our mission is to meticulously evaluate the expression ${ \frac{\cos A (1 + \cos A)}{\cos B (1 - \sin B)} }$. This exploration will not only test our understanding of fundamental trigonometric identities but also hone our problem-solving skills in manipulating these identities to arrive at a concise solution. Trigonometry, at its core, is the study of relationships between the sides and angles of triangles. These relationships are expressed through trigonometric functions such as sine, cosine, tangent, and their reciprocals. Mastering these functions and their interdependencies is crucial for tackling a wide range of mathematical problems, from simple triangle calculations to complex engineering and physics applications. This particular problem ingeniously combines the concepts of complementary angles and trigonometric identities, offering a rich context for deepening our understanding of these principles. The given condition, ${ \hat{A} + \hat{B} = 90^\circ }$, immediately tells us that angles A and B are complementary. This relationship is a cornerstone of trigonometric identities, as it implies specific connections between the sine and cosine of these angles. Specifically, the sine of an angle is equal to the cosine of its complement, and vice versa. This fundamental relationship will play a pivotal role in simplifying the expression we aim to evaluate. The second piece of information provided, ${ \sin A = \frac{11}{61} }$, gives us a concrete numerical value to work with. By leveraging trigonometric identities, we can use this value to determine other trigonometric functions of angle A, such as cosine and tangent. These derived values will then be instrumental in simplifying the overall expression. The expression ${ \frac{\cos A (1 + \cos A)}{\cos B (1 - \sin B)} }$ appears complex at first glance. However, by strategically applying trigonometric identities and the relationship between complementary angles, we can systematically reduce it to a simpler form. This process will involve algebraic manipulation, careful substitution, and a keen eye for recognizing opportunities for simplification. Ultimately, the goal is to arrive at a single numerical value that represents the expression's evaluation. This journey through trigonometric identities and problem-solving techniques will not only provide the answer to this specific problem but also enhance our broader mathematical toolkit. So, let us embark on this exploration, armed with our knowledge of trigonometry and a determination to unravel the solution.

The complementary angle relationships are pivotal in simplifying trigonometric expressions, especially when dealing with angles that sum up to 90 degrees. In our case, given that ${ A + B = 90^\circ }$, we can immediately establish the fundamental relationships between the sine and cosine of angles A and B. These relationships form the bedrock of our simplification strategy. The core concept of complementary angles hinges on the fact that the sine of an angle is equal to the cosine of its complement, and vice versa. Mathematically, this can be expressed as ${ \sin A = \cos B }$ and ${ \cos A = \sin B }$. These identities are not merely abstract formulas; they stem directly from the geometric properties of right-angled triangles. In a right-angled triangle, the two acute angles are complementary, and the sine of one acute angle is defined as the ratio of the opposite side to the hypotenuse, while the cosine of the other acute angle is the ratio of the adjacent side to the hypotenuse. These ratios are inherently equal, giving rise to the complementary angle identities. In the context of our problem, these relationships offer a powerful tool for transforming the expression we need to evaluate. By substituting ${ \cos B }$ with ${ \sin A }$ and ${ \sin B }$ with ${ \cos A }$, we can rewrite the expression in terms of a single angle, A, which simplifies the subsequent calculations significantly. This substitution is not just a superficial change; it fundamentally alters the structure of the expression, making it more amenable to further simplification. The strategic use of complementary angle relationships is a hallmark of efficient trigonometric problem-solving. It allows us to bridge the gap between seemingly disparate parts of an expression, revealing underlying connections and paving the way for a more elegant solution. By recognizing and applying these relationships, we can avoid unnecessary complications and focus on the core elements of the problem. In our specific case, the substitution of ${ \cos B }$ with ${ \sin A }$ is a crucial step in untangling the expression ${ \frac{\cos A (1 + \cos A)}{\cos B (1 - \sin B)} }$. This substitution allows us to express the entire expression in terms of ${ \sin A }$ and ${ \cos A }$, which are directly related to the given information, ${ \sin A = \frac{11}{61} }$. This strategic manipulation sets the stage for the next phase of our problem-solving journey, where we will leverage the given sine value and trigonometric identities to determine the value of ${ \cos A }$ and ultimately evaluate the entire expression. The power of complementary angle relationships lies in their ability to connect seemingly distinct trigonometric functions, providing a crucial link for simplification and problem-solving. As we move forward, we will see how this fundamental concept, combined with other trigonometric identities, leads us to a clear and concise solution.

To determine the value of ${ \cos A }$, we can leverage the fundamental Pythagorean trigonometric identity, which states that ${ \sin^2 A + \cos^2 A = 1 }$. This identity is a cornerstone of trigonometry and provides a direct relationship between the sine and cosine of any angle. Given that we know ${ \sin A = \frac{11}{61} }$, we can substitute this value into the Pythagorean identity and solve for ${ \cos A }$. This process involves algebraic manipulation and a careful consideration of the possible solutions. The Pythagorean identity is not just an abstract formula; it arises directly from the Pythagorean theorem applied to a right-angled triangle. In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. By defining the trigonometric functions in terms of the ratios of these sides, we arrive at the identity ${ \sin^2 A + \cos^2 A = 1 }$. This geometric foundation gives the identity its inherent validity and makes it a powerful tool for relating sine and cosine values. Substituting ${ \sin A = \frac{11}{61} }$ into the Pythagorean identity, we get ${ \left(\frac{11}{61}\right)^2 + \cos^2 A = 1 }$. This equation can be rearranged to isolate ${ \cos^2 A }$, giving us ${ \cos^2 A = 1 - \left(\frac{11}{61}\right)^2 }$. Evaluating the square of the fraction, we have ${ \cos^2 A = 1 - \frac{121}{3721} }$. To combine these terms, we need a common denominator, which is 3721. Thus, ${ \cos^2 A = \frac{3721}{3721} - \frac{121}{3721} = \frac{3600}{3721} }$. Now, to find ${ \cos A }$, we take the square root of both sides: ${ \cos A = \pm \sqrt{\frac{3600}{3721}} }$. The square root of 3600 is 60, and the square root of 3721 is 61. Therefore, ${ \cos A = \pm \frac{60}{61} }$. However, we need to consider the sign of ${ \cos A }$. Since we are not given any specific information about the quadrant in which angle A lies, we must consider both positive and negative possibilities. However, in typical trigonometric problems involving triangles, we often assume that angles are acute (less than 90 degrees), in which case cosine is positive. Therefore, we will take the positive value, ${ \cos A = \frac{60}{61} }$. This determination of ${ \cos A }$ is a crucial step in our problem-solving process. With both ${ \sin A }$ and ${ \cos A }$ known, we are now equipped to tackle the evaluation of the original expression. The Pythagorean identity has served as a bridge, connecting the given sine value to the corresponding cosine value, and empowering us to move closer to the final solution. The careful application of this identity, combined with algebraic manipulation, has yielded a key piece of the puzzle, setting the stage for the final evaluation.

Now, with the values of both ${ \sin A }$ and ${ \cos A }$ determined, we can proceed to substitute these values into the original expression and embark on the simplification process. This is a crucial step where we bring together all the pieces of information we have gathered and manipulate them algebraically to arrive at the final answer. The original expression we are tasked with evaluating is ${ \frac{\cos A (1 + \cos A)}{\cos B (1 - \sin B)} }$. As we established earlier, using the complementary angle relationships, we can replace ${ \cos B }$ with ${ \sin A }$ and ${ \sin B }$ with ${ \cos A }$. This substitution transforms the expression into ${ \frac{\cos A (1 + \cos A)}{\sin A (1 - \cos A)} }$. This transformation is a significant simplification, as it now expresses the entire expression in terms of ${ \sin A }$ and ${ \cos A }$, both of which we know. We have ${ \sin A = \frac{11}{61} }$ and ${ \cos A = \frac{60}{61} }$. Substituting these values into the transformed expression, we get ${ \frac{\frac{60}{61} \left(1 + \frac{60}{61}\right)}{\frac{11}{61} \left(1 - \frac{60}{61}\right)} }$. This looks complex, but we can systematically simplify it step by step. First, let's simplify the terms inside the parentheses: ${ 1 + \frac{60}{61} = \frac{61}{61} + \frac{60}{61} = \frac{121}{61} }$ and ${ 1 - \frac{60}{61} = \frac{61}{61} - \frac{60}{61} = \frac{1}{61} }$. Now, our expression becomes ${ \frac{\frac{60}{61} \cdot \frac{121}{61}}{\frac{11}{61} \cdot \frac{1}{61}} }$. To simplify this further, we can multiply the fractions in the numerator and the denominator: Numerator: ${ \frac{60}{61} \cdot \frac{121}{61} = \frac{60 \cdot 121}{61 \cdot 61} = \frac{7260}{3721} }$ Denominator: ${ \frac{11}{61} \cdot \frac{1}{61} = \frac{11 \cdot 1}{61 \cdot 61} = \frac{11}{3721} }$ Now, the expression is ${ \frac{\frac{7260}{3721}}{\frac{11}{3721}} }$. To divide fractions, we multiply by the reciprocal of the denominator: ${ \frac{\frac{7260}{3721}}{\frac{11}{3721}} = \frac{7260}{3721} \cdot \frac{3721}{11} }$. We can cancel out the common factor of 3721: ${ \frac{7260}{3721} \cdot \frac{3721}{11} = \frac{7260}{11} }$. Finally, we divide 7260 by 11: ${ \frac{7260}{11} = 660 }$. Therefore, the value of the expression ${ \frac{\cos A (1 + \cos A)}{\cos B (1 - \sin B)} }$ is 660. This process of substitution and simplification highlights the power of algebraic manipulation in conjunction with trigonometric identities. By carefully substituting the known values and systematically simplifying the expression, we have arrived at a concise numerical answer. The journey from the initial complex expression to the final answer of 660 demonstrates the elegance and effectiveness of trigonometric problem-solving techniques.

In conclusion, we have successfully evaluated the trigonometric expression ${ \frac{\cos A (1 + \cos A)}{\cos B (1 - \sin B)} }$ given that ${ A + B = 90^\circ }$ and ${ \sin A = \frac{11}{61} }$. The final result of this evaluation is 660. This problem provided a comprehensive exercise in applying fundamental trigonometric concepts and problem-solving techniques. We began by recognizing the significance of the complementary angle relationship ${ A + B = 90^\circ }$, which allowed us to establish the key identities ${ \sin A = \cos B }$ and ${ \cos A = \sin B }$. These identities served as a bridge, connecting the trigonometric functions of angles A and B and enabling us to simplify the expression. Next, we utilized the Pythagorean identity, ${ \sin^2 A + \cos^2 A = 1 }$, to determine the value of ${ \cos A }$ given the value of ${ \sin A }$. This step was crucial, as it provided us with the necessary information to substitute numerical values into the expression. The algebraic manipulation involved in solving for ${ \cos A }$ demonstrated the importance of mastering fundamental algebraic skills in the context of trigonometric problem-solving. With both ${ \sin A }$ and ${ \cos A }$ known, we proceeded to substitute these values into the transformed expression. This step required careful attention to detail and a systematic approach to avoid errors in calculation. The subsequent simplification process involved a series of algebraic manipulations, including combining fractions, multiplying terms, and canceling common factors. Each step in this process built upon the previous one, gradually leading us closer to the final answer. The final simplification involved dividing 7260 by 11, which yielded the result of 660. This numerical answer represents the value of the original trigonometric expression under the given conditions. The entire problem-solving process, from the initial recognition of complementary angle relationships to the final calculation, underscores the interconnectedness of trigonometric concepts. The Pythagorean identity, complementary angle relationships, and algebraic manipulation techniques all played vital roles in arriving at the solution. This exercise has not only provided us with a specific answer but has also reinforced our understanding of fundamental trigonometric principles and enhanced our problem-solving abilities. The ability to effectively manipulate trigonometric expressions is essential for a wide range of applications, from engineering and physics to computer graphics and navigation. By mastering these techniques, we equip ourselves with a powerful toolset for tackling complex mathematical problems in various fields. In conclusion, the evaluation of the expression ${ \frac{\cos A (1 + \cos A)}{\cos B (1 - \sin B)} }$ to 660 serves as a testament to the power and elegance of trigonometric problem-solving. The journey from the initial problem statement to the final answer has been a valuable exercise in applying fundamental concepts and honing our mathematical skills.