Question

Question #5a859

Answer 1

We will start from the left hand side and show that it equals the right hand side. To do so, we will use the following identities:

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

Note that the third identity may be derived from `sin^2(x) + cos^2(x) = 1` by dividing each side by `cos^2(x)` and then subtracting `1` from each side.

Proceeding:

`LHS = tan^2(u) - sin^2(u)`

`= (sin(u)/cos(u))^2-sin^2(u)`

`= sin^2(u)/cos^2(u)-sin^2(u)`

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

`=sin^2(u)(sec^2(u)-1)`

`=sin^2(u)tan^2(u) = RHS` ∎