Question

How do you prove `sec^2 x - cot^2 ( pi/2-x) =1`?

Answer 1

Using the following:

• `sec(x) = 1/cos(x)`
• `cot(x) = cos(x)/sin(x)`
• `cos(-x) = cos(x)`
• `sin(-x) = -sin(x)`
• `cos(x-pi/2) = sin(x)`
• `sin(x-pi/2) = -cos(x)`
• `sin^2(x)+cos^2(x) = 1 => 1 - sin^2(x) = cos^2(x)`

We have

`sec^2(x)-cot^2(pi/2-x) = sec^2(x) - cot^2(-(x-pi/2))`

`=1/cos^2(x) - cos^2(-(x-pi/2))/sin^2(-(x-pi/2))`

`=1/cos^2(x)-cos^2(x-pi/2)/(-sin(x-pi/2))`

`=1/cos^2(x) - sin^2(x)/cos^2(x)`

`= (1-sin^2(x))/cos^2(x)`

`=cos^2(x)/cos^2(x)`

`= 1`