If \( y = \tan^{-1}(x) + \tan^{-1}(y) \), what is the value of \( y \) when \( x

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Question: If \\( y = \\tan^{-1}(x) + \\tan^{-1}(y) \\), what is the value of \\( y \\) when \\( x = 1 \\)?

Options:

Correct Answer: \\( \\frac{\\pi}{4} \\)

Solution:

When \\( x = 1 \\), \\( y = \\tan^{-1}(1) + \\tan^{-1}(y) \\) leads to \\( y = \\frac{\\pi}{4} \\).

If \( y = \tan^{-1}(x) + \tan^{-1}(y) \), what is the value of \( y \) when \( x = 1 \)?

If \( y = \tan^{-1}(x) + \tan^{-1}(y) \), what is the value of \( y \) when \( x = 1 \)?

Step 1: Start with the equation given in the question: y = tan^(-1)(x) + tan^(-1)(y).
Step 2: Substitute x = 1 into the equation: y = tan^(-1)(1) + tan^(-1)(y).
Step 3: Calculate tan^(-1)(1). The value of tan^(-1)(1) is π/4 because the tangent of π/4 is 1.
Step 4: Now the equation becomes: y = π/4 + tan^(-1)(y).
Step 5: To solve for y, we can use the fact that tan^(-1)(y) is the angle whose tangent is y. This means we can express y in terms of tangent: y = π/4 + tan^(-1)(y).
Step 6: We can assume y = k, where k is some value we want to find. So we rewrite the equation: k = π/4 + tan^(-1)(k).
Step 7: We can find the value of k by recognizing that if k = π/4, then tan^(-1)(k) = tan^(-1)(π/4) = π/4.
Step 8: Substitute k = π/4 back into the equation: π/4 = π/4 + tan^(-1)(π/4).
Step 9: Since both sides are equal, we conclude that y = π/4 is a valid solution.

Inverse Trigonometric Functions – Understanding the properties and values of inverse tangent functions.
Implicit Equations – Solving equations where the variable appears on both sides.