Inverse Trigonometric Functions

We are given the function:

To find , we need to find the inverse of the function and then substitute for .

Let .

Now solve for in terms of :

This is the inverse function:

Now substitute into the inverse function:

Answer:

Given expression:

Using the identity:

Let , so .

Applying the identity:

Calculating the denominator:

Calculating the numerator:

Dividing:

Thus, the correct answer is:

Given:

Let , so we have:

Using the Pythagorean identity:

Substituting :

Thus, the correct answer is:

Given expression:

Using the identity:

, when

Here, and .

Calculating :

Calculating :

Using the identity:

Thus, the correct answer is:

We are given , which means we need to find the angle such that We need to calculate . Using the identity for sine and cosine in a right triangle, if , then the opposite side is 7 and the hypotenuse is 25. We can find the adjacent side using the Pythagorean theorem: Thus, Therefore,

The correct option is (C) :

Step 1: Evaluate the inner cosine function:

Step 2: Apply the inverse cosine:

Step 1: Evaluate . Recall that , so:

Step 2: Evaluate . Let , then . This implies , so:

Step 3: Subtract the two results:

Conclusion: The correct answer is .

Step 1: Apply the tangent addition formula for inverse functions: Here, and , so .

Step 2: Substitute the values:

Step 3: Simplify the expression:

Conclusion: The exact value is .

We are given that: Since gives the angle whose tangent is , the above equation implies:
Thus, the value of is .