Question

tan^2 x/(secx-1)=(1-cosx)/(1+cosx)

Answer 1

tan(x) = sin(x)/cos(x) 
sec(x) = 1/cos(x) 
sin^2(x) = 1 - cos^2(x) 
LHS: 
tan^2(x)/(sec(x) + 1) 
tan^2(x)/(1/cos(x) + 1) 
tan^2(x)/[(1 + cos(x)/cos(x)] 
tan^2(x) * cos(x)/(1 + cos(x) 

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

(1 - cos(x))(1 + cos(x))/[cos(x)*(1 + cos(x))] 

(1 - cos(x))/cos(x) 

LHS = RHS 

Proved

Remember the identity tan^2 x = sec^2 x - 1 [derivation: cos^2 + sin^2 = 1 from the Pythagorean theorem; divide by cos^2 to get 1 + tan^2 = sec^2] Also... show more