Question Type 3: Rewriting trigonometric equations using the double angle identities and golden rule Exercises

Derive the identity for $\sin(2\theta)$ in terms of $\tan\theta$. Present the final expression as a function of $\tan\theta$.

Prove that $\cos(3\theta)=4\cos^3\theta-3\cos\theta.$

Rewrite the product $\sin\theta\cos(3\theta)$ as a sum or difference of trigonometric functions using product-to-sum identities.

Using double-angle identities, derive an expression for $\sin(4\theta)$ in terms of $\sin\theta$ and $\cos\theta$.

Express $\sin^2\theta\,\cos^2\theta$ in terms of $\cos(4\theta)$ and constants.

Express $\cos(2\theta)$ using only $\sin\theta$.

Express $\cos(4\theta)$ using only $\sin\theta$. Simplify your result.

Derive the identity for $\tan(2\theta)$ in terms of $\tan\theta$ only.

Express $\cos(4\theta)$ using only $\cos\theta$. Provide the simplified polynomial form in $\cos\theta$.

Derive the double-angle identity for $\cos(2\theta)$ in terms of $\cos\theta$. Provide a final expression involving only $\cos\theta$.