1/(1 - sec(theta)) + 1/(1 + sec(theta)) = - 2cot^2 theta

Question

verify the identity

Answer 1

Doing the left side (general practice is to operate only on one side)

Adding using LCD (1 - sec θ)(1 + sec θ) and adding

(1 + sec θ + 1 - sec θ) / 1 - sec² θ)

2 / (1 - sec² θ)

2 / tan² θ

2 cot² θ

Note: -2 in post is incorrect.

Answer 2

1/(1 - sec(theta)) + 1/(1 + sec(theta)) = - 2cot^2 theta

sometimes I use this technique, I say cos(theta)=x, sin(theta)=y in the unit circle

and then remember x²+y²=1 (unit circle)

1/(1-1/x)+1/(1+1/x)=-2(x/y)²

x/(x-1)+x(x+1)=-2(x/y)²

[x^2-x+x²+x]/(x²-1)=-2(x/y)²

2x²/-y²=-2(x/y)²,,,,,,where x²-1=-y²