( sin ( x ) ) ( tan ( x ) ⋅ cos ( x ) − cot ( x ) ⋅ cos ( x ) ) = 1 − 2 cos ^2 x

Question

I don't know how to verify this identity. Can you explain it to me?

Answer 1

Work with the following 3 properties, substitute, and simplify:

 

tan(x) = sin(x)/cos(x)

cot(x) = cos(x)/sin(x)

sin²(x) = 1 - cos²(x)

 

sin(x) [tan(x)cos(x) − cot(x)cos(x)] = sin(x) {[sin(x)/cos(x)]cos(x) − [cos(x)/sin(x)]cos(x)}

 

= sin(x) [sin(x) - cos²(x)/sin(x)] = sin²(x) - cos²(x) = [1 - cos²(x)] - cos²(x) = 1 - 2cos²(x)