2(sinx-sin^3 x)/cosx= sin2x

Question

verify the identity 

Answer 1

Before starting, it's important to know the Pythagorean identity tells us:

 

sin²x + cos²x = 1

cos²x = 1 - sin²x

 

Also note the double angle identity for sin

 

sin(2x) = 2sinxcosx

 

These identity will be used during your problem. Now we'll begin.

 

2(sinx - sin³x)/cosx = sin2x

2sinx(1 - sin²x)/cosx

2sinx(cos²x)/cosx

2sinxcosx

sin2x

 

In step 2, I factored out 1 sinx. In step 3 I used the Pythagorean identiy. In step 4 I divided out the cosx which was on the bottom of the fraction. I hope this helps.