for what values of x is the following expression defined?

Question

tan(x)/1-2cos^2 x

This question has given me a brain fart and cant figure it out. If anyone could help it would be greatly appreciated! Thanks!

Answer 1

The only time it's undefined is when the denominator equals 0.

Rewrite tan x as sin x/ cos x.

Therefore the function becomes: sin x / (cos x (1 - 2cos²x))

Therefore cos x(1-2cos²x )≠ 0

We'll solve for x.

Therefore we got 2 cases:

Case 1: cos x ≠ 0

cos x = 0 when x = π/2 + πn

Therefore it's undefined whenever x = π/2 + πn

Case 2: 1-2cos²x ≠ 0

1 ≠ 2cos²x

1/2 ≠cos²x

√2/2 ≠ cos x

cos x = √2/2 when x = π/4 + 2πn and x = -π/4 + 2πn.

Therefore it's also undefined whenever x = π/4 + 2πn and x = -π/4 + 2πn.

Answer 2

1 - 2 cos² x > 0

-2 cos² x > -1

cos² x > 0.5

cos x > 0.7071

x > 45°