Trig Identities

Question

Prove the identity: tan^2(x)-sin^2(x)= tan^2(x)sin^2(x)

Answer 1

Start with the left hand side

 

tan^2(x) - sin^2(x)

 

Replace tan(x) with sin(x)/cos(x)

 

sin^2(x) / cos^2(x)  -  sin^2(x)

 

Put both terms over the common denominator cos^2(x)

 

[ sin^2(x) - sin^2(x)cos^2(x) ] / cos^2(x)

 

Factor out sin^2(x) from the numerator

 

sin^2(x) [ 1 - cos^2(x) ] / cos^2(x)

 

Use the identity 1 - cos^2(x) = sin^2(x) and convert sin^2(x)/cos^2(x) back to tan^2(x)

 

tan^2(x)sin^2(x)

 

Which is the right hand side