Trigonometry & Inverse Trigonometry Question - None

Question

Ex. 35. Statement I: If tan⁻¹x + tan⁻¹y + tan⁻¹z = π/4 and x + y + z = 1, then arithmetic mean of odd powers of x, y, z is equal to 1/3.

Statement II: For any x, y, z we have xyz − xy − yz − zx + x + y + z = 1 + (x − 1)(y − 1)(z − 1)

Accepted Answer

Solution:

Let x = tan A, y = tan B, z = tan C.

Then A + B + C = π/4

Using the formula: $ an(A + B + C) = \frac{x + y + z - xyz}{1 - (xy + yz + zx)}$

Since tan(π/4) = 1:

$1 = \frac{x + y + z - xyz}{1 - (xy + yz + zx)}$

⟹ 1 − (xy + yz + zx) = x + y + z − xyz

⟹ (x − 1)(y − 1)(z − 1) = 0

⟹ One of x, y, z equals 1.

If z = 1, then x + y = 0, so y = −x.

Odd powers: x^(odd) + (−x)^(odd) + 1^(odd) = x − x + 1 = 1

AM of odd powers = 1/3

Statement I is true and Statement II is true. Statement II is an algebraic identity that holds independently and does not directly explain why the AM of odd powers equals 1/3.

∴ Answer is (b).

Question:

Step-by-Step Solution

Master Trigonometry & Inverse Trigonometry with Mathbee