Inverse Trigonometric Functions

We are given the function Recall that the sum of inverse sine and inverse cosine of complementary angles is , i.e. By manipulating the given expression and using this identity, we can see that the values of and must add up to . Therefore, the range of is a constant value, specifically .

We are given the equation: Step 1: Use the identity for the sum of arctangents The identity for the sum of arctangents is: Using this identity, we can write: Step 2: Set the equation equal to We know that , so we set the argument of the arctangent equal to 1: Multiply both sides by : Step 3: Solve the quadratic equation Rearrange the terms: Solve this quadratic equation using the quadratic formula: where . Substituting these values: Thus, the real values of are .

We are given the expression: We need to simplify and solve this expression. Step 1: Simplifying The expression is the inverse cotangent of 4. We can write this as: Thus, we know . Step 2: Solving Next, we are given . We can write this as: Thus, we know . Step 3: Combining the expressions Now, combine the two expressions and simplify the result. Using standard trigonometric identities, we find that the simplified result of the expression is . Thus, the correct answer is .

Given: Similarly, So:

To find the derivative of with respect to at , where , we will use the derivatives of inverse tangent functions and sum rule of differentiation. The derivative of with respect to is .

Let and . We differentiate each:

For :

Substituting back at , .

Thus, .

For :

Substituting back at , .

Thus, .

Then, combine derivatives for both inverse tangent functions using the sum rule:
.

Therefore, the value of at is 10.

Use identity: Group and simplify: Step 1: Repeat and finally reduce: All simplify to: