Trigonometry and Modelling

A student is asked to derive several double-angle identities for a classroom presentation using the compound angle formulae.

By substituting $B = x$ into the addition formula for $sin⁡(x+B)\sin(x + B)$ , show that $sin⁡2x=2sin⁡xcos⁡x\sin 2x = 2 \sin x \cos x$ .

[2]

Using the identity for $cos⁡(A+B)\cos(A + B)$ , derive an expression for $\cos 2x\,$ in terms of $\sin x\,$ and $cos⁡x\cos x$ .

[2]

ci.

Hence, show that $cos 2x = 2 \cos^2 x - 1$ .

[2]

cii.

Hence, show that $cos 2x = 1 - 2 \sin^2 x$ .

[2]

Use the formula for $tan⁡(A+B)\tan(A + B)$ to derive the identity for $\tan 2x\,$ in terms of $tan⁡x\tan x$ .

[2]

Trigonometry and Modelling Questions