IB Mathematics Analysis and Approaches Question Type 2: Simple Proofs with the Golden Rule Exercises

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

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 ${\sec }^{2}x-{\tan }^{2}x=1$.