Question Type 2: Simple proofs with the golden rule Exercises

Prove that $\operatorname{cosec}^2 x - \cot^2 x = 1$.

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

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

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

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

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

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

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

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.

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

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

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