Pre Calc: Identities

Question

Find all solutions to the equation on [0,2π)

sinxcotx^2-sinxcsc^2 x=1/2

Answer 1

Rewrite in terms of sine and cosine.

 

sinx(cosx / sinx)² - (sinx / sin²x) = 1/2

 

(cos²x / sinx) - (1 / sinx) = 1/2

 

(cos²x - 1) / sinx = 1 / 2

 

 

Cross-multiply.

 

2(cos²x - 1) = sinx

 

2cos²x - 2 - sinx = 0

 

2(1 - sin²x) - 2 - sinx = 0

 

2 - 2sin²x - 2 - sinx = 0

 

0 = 2sin²x + sinx

 

0 = sinx(2sinx + 1)

 

 

Set the factors equal to zero and solve for x.