Question Type 2: Simple proofs with the golden rule Bootcamps

Simplify the expression $\tan^2 x + 1 - \sec^2 x$.

Prove $1 - \sin^2 x = \cos^2 x$.

Prove the identity $1 + \cot^2 x = \,csc^2 x$.

Prove the Pythagorean identity $1 + \tan^2 x = \,sec^2 x$.

Derive the identity $\tan^2 x = \sec^2 x - 1$.

Show that $\sec^2 x - \tan^2 x = 1$.

Using the Pythagorean identity, prove that $\frac{1}{\cos^2 x} - \tan^2 x = 1$.

Show that $1 + \tan^2 x = \frac{\sin^2 x + \cos^2 x}{\cos^2 x}$ and hence simplify to a single trigonometric function.

Prove that $\csc^2 x - \cot^2 x = 1$.

Given $\tan x = \frac{a}{b}$ for constants $a,b$, express $\sec x$ in terms of $a$ and $b$.

Prove that $(\sec x + \tan x)(\sec x - \tan x) = 1$.

Derive the tangent addition formula $\tan(x+y)=\frac{\tan x + \tan y}{1 - \tan x\tan y}$ by expressing in terms of sine and cosine.