State the sine addition formula for \(\sin(A+B)\).

State the cosine addition formula for \(\cos(A+B)\).

Use an addition formula to expand \(\sin(x+\frac\pi6)\).

Use a difference formula to expand \(\cos(x-\frac\pi3)\).

Use exact values to find \(\sin\left(\frac\pi4+\frac\pi6\right)\).

Use exact values to find \(\cos\left(\frac\pi4+\frac\pi6\right)\).

Derive the double-angle formula for \(\sin(2A)\) from the sine addition formula.

Use a double-angle formula to find \(\sin(2x)\) if \(\sin x=\frac35\) and \(\cos x=\frac45\).

Use the tangent addition formula to simplify \(\tan(A+B)\) when \(\tan A=2\) and \(\tan B=\frac13\).

Use the cosine double-angle formula to find \(\cos(2x)\) if \(\cos x=\frac{5}{13}\) and \(\sin x=\frac{12}{13}\).

Show that \(\cos(2A)=2\cos^2A-1\).

Show that \(\cos(2A)=1-2\sin^2A\).

Find \(\tan(2x)\) if \(\tan x=\frac12\).

Find \(\cos\left(\frac\pi{12}\right)\) by writing \(\frac\pi{12}=\frac\pi4-\frac\pi6\).

Use a half-angle formula to find \(\sin\left(\frac\pi8\right)\).

For which values of \(A\) is the formula \(\tan(2A)=\frac{2\tan A}{1-\tanan^2A}\) undefined because of its denominator?

A student writes \(\cos(A+B)=\cos A\cos B+\sin A\sin B\). Diagnose the mistake using \(A=B=\frac\pi4\).

Prove the identity \(\sin(A+B)+\sin(A-B)=2\sin A\cos B\).

Prove the identity \(\cos(A-B)-\cos(A+B)=2\sin A\sin B\).

The displacement of a simple oscillator is \(x(t)=R\cos(\omega t+\phi)\). Expand it into a sum involving \(\cos(\omega t)\) and \(\sin(\omega t)\), and identify the coefficients.