cot^2x-csc x-1=0

Answer 1

Rewrite in terms of sin and cos to get:

cos²(x)/sin²(x) - 1/sin(x) - 1 = 0

Multiply through by sin²(x) to get:

cos²(x) - sin(x) - sin²(x) = 0

Using the Pythagorean Identity cos²(x) = 1 - sin²(x), replace cos²(x) to get:

1 - sin²(x) - sin(x) - sin²(x) = 0

Combine like terms to get:

-2sin²(x) - sin(x) + 1 = 0

Multiply by -1 to get:

2sin²(x) + sin(x) - 1 = 0

Make the substitution w = sin(x) to get:

2w² + w - 1 = 0

Factor to solve the quadratic:

(2w -1)(w + 1) = 0

w = 1/2 and w = -1

Substitute back (w = sin(x)) to get:

sin(x) = 1/2 and sin(x) = -1

x = π/6, 5π/6, 3π/2

If the domain is restricted to 0 < x < 2π this will be the answer, if not, you'll need to make these generic so x = π/6 + 2πk, 5π/6 + 2πk, 3π/2 + 2πk