Trigonometry

We are asked to simplify the expression: Step 1: Rewrite in terms of sine and cosine We know the following identities: Substitute these into the original expression: Step 2: Simplify the expression To simplify the numerator: For the denominator: Now, the expression becomes: Step 3: Recognize the result Simplifying the expression leads us to the form , which is the correct answer.

We are given the following two equations:
1.
2. We need to find . Step 1: Use the identity for We know that: Step 2: Square and add the two given equations First, square both equations: Now, add the two squared equations: Step 3: Simplify Using the Pythagorean identity , the equation simplifies to: Step 4: Solve for Simplify the equation: Since , we can use this result to find: Thus, the correct answer is .

We are tasked with evaluating the expression: First, let's simplify each part individually.
Step 1: Simplifying
The principal value of the inverse cosine function, , lies between and . Since is greater than , we need to find an equivalent angle within the range by subtracting multiples of . Now, since for , we have:
Step 2: Simplifying
The principal value of the inverse sine function, , lies between and . Since is greater than , we again find an equivalent angle within the range by subtracting multiples of . Now, since for , we have:
Step 3: Subtracting the Results
Now, we subtract the two results: Thus, the value of the given expression is .

We know that: Using the formula for , where and : Substitute the known values of , , and : Thus: