Trigonometric Identities

We simplify the given expression: Using the identity , we can rewrite the expression as: Thus, the correct answer is .

Using the odd property of the sine function, we know that: Thus, the correct answer is .

We know that . Therefore, . Substitute this into the given expression: Thus, the correct answer is .

Using the identity and , we can simplify the expression and find that the value of . Thus, the correct answer is 0.

Using the sum of trigonometric identities, we can simplify: Thus, the correct answer is .

We start by expanding the trigonometric expression , , and . Using trigonometric identities for and : Now, substituting into the given expression, it does not simplify directly to any of the options provided. Therefore, the correct answer is (4) None of these.

We are given that . Rearranging this equation: Now, using the identity , we substitute for : Now, calculate : Simplifying, we find that this expression equals . Therefore, the correct answer is .

We are given that: Now, is a difference of squares, so we can use the identity : Simplifying: Thus, the correct answer is .

We can use the identity for tangent addition: In this case, we have: Thus, the correct answer is -1.

Use the formula for the cosine of the difference of two angles: Substitute the values of and , and calculate and . After solving, we find: Thus, the correct answer is .

We know that , which implies and . Similarly, , which implies and . Now, calculate: Thus, the correct answer is 0.

To find , we follow these steps:

1. Given: .
2. Square both sides to use trigonometric identities:
3. Expand the left-hand side:
4. Use the identity :
5. Solve for :
6. Match with options: corresponds to option (B).

Thus, the correct answer is: