Trigonometry

We are given: We need to find . We will use the formula for the tangent of the sum of two angles: Substitute the given values for and : First, simplify the numerator: Now, simplify the denominator: So, The value of corresponds to an angle of . Thus, .

We are given the equation: The sum of sines reaches its maximum value when each sine term is equal to 1, which occurs when each angle is . Hence, we assume: For , we have: Thus: Now, for the cosine terms: So:

We are given that . First, let's calculate and for . Step 1: Calculate for : - The tangent of is: Since the angle is in the third quadrant, is positive: So, Step 2: Calculate for : - The secant is the reciprocal of cosine, and the cosine of is: So, Step 3: Now, calculate : Thus, the value of is 1.