Trigonometric Identities

We have,

Step 1: Use the Addition Formula for Tangents
We are given: We can use the addition formula for tangents:
Step 2: Substitute the Values
Substituting the given values: Simplify the numerator and denominator: Thus, .
Step 3: Conclusion Thus, .

We are given the expression:
Step 1: Use the formula for the difference of arctangents
We use the standard identity for the difference of two inverse tangents: Let and .
Applying the identity:
Step 2: Simplify the expression
After simplification, the result simplifies to , which is the required value.
Step 3: Conclusion
Thus, the correct answer is , which corresponds to option (a).

We are given the equation:
Step 1: Use of Symmetry
The equation suggests that the values of and are related in such a way that symmetry can be applied.
Specifically, this indicates that and may have specific values at certain points for which the equation holds true.
For instance, .
Step 2: Evaluate Trigonometric Identity
For , we know: This simplifies the equation, and both sides match.
Hence, for , the value of becomes:
Step 3: Conclusion
Thus, the correct answer is , which corresponds to option (a).