Practice section 4 of 8

If $x=2\tan\theta$, where $0^\circ<\theta<90^\circ$, show that $\dfrac{x}{\sqrt{4+x^2}}=\sin\theta$.

Prove that $\cos3\theta\equiv4\cos^3\theta-3\cos\theta$. The formulae for $\sin2\theta$ and $\cos2\theta$ may be quoted without proof.

Show that $\cos^4\theta\equiv\dfrac18(3+4\cos2\theta+\cos4\theta)$.

Find the maximum and minimum values of $2\cos x-3\sin x$.

Express $5\sin\theta+12\cos\theta$ in the form $r\cos(\theta-\alpha)$. Show that $-6\le5\sin\theta+12\cos\theta+7\le20$.

Differentiate $\dfrac1{x^3}-x^5+\sqrt{x^3}$ with respect to $x$.