Question Type 1: Rewriting values of sin and cos using the golden rule Exercises

Evaluate $\displaystyle \int_{0}^{\frac{\pi}{2}}5\cos^2 x\,dx$ by rewriting the integrand.

Simplify $\sin^2 x \cos^2 x$ in terms of $\cos(4x)$.

Express $5\cos^2 x$ in terms of $\sin x$.

Show that $1 - \sin^2 x = \cos^2 x$ and use this to express $8 - 8\sin^2 x$ in simplest form.

Simplify $\cos^4 x - \sin^4 x$ and express your answer in terms of $\cos(2x)$.

Express $\sin^4 x$ in the form $A + B\cos(2x) + C\cos(4x)$, and state the values of $A$, $B$, and $C$.

Rewrite $3\sin^2 x + 7\cos^2 x$ solely in terms of $\sin^2 x$.

Express $5\sin^2 x - 3\cos^2 x$ in the form $D + E\cos(2x)$.

Simplify $\cos^2 x - \sin^2 x + 2\sin x\cos x$ in terms of $\sin(2x)$ or $\cos(2x)$.

Rewrite $3\sin x + 4\cos x$ in the form $R\sin\bigl(x+\alpha\bigr)$, where $R>0$ and $0<\alpha<\tfrac{\pi}{2}$.

Express $\sin^2 x - \sin^4 x$ in terms of $\cos(4x)$.

Simplify $\cos^4 x + \sin^4 x$ in the form $F + G\cos(4x)$.