show that 1/(1-cos x) - 1/(1+cos x)=2cosec x cot x

Question

show that 1/(1-cos x) - 1/(1+cos x)=2cosec x cot x

Answer 1

1/(1-cos x) - 1/(1+cos x)=2cosec x cot x

 

consider LHS (left hand side) of problem

 

1/(1-cos x) - 1/(1+cos x)

Lcm of denominator 1² - cos²x

 

[(1+cos x) -(1-cos x)] / 1²-cos²x

 

(1+cos x-1+cos x)/ sin²x

 

2 cos x / sin² x

 

2 cos x / sin x (sin x)

 

2( cos x /sin x ) *   (1 * sin x)

 

2 cot x             * cosecx

 

2 cosec x cot x