IB Mathematics Analysis and Approaches Question Type 1: Rewriting Values of Sin and Cos Using the Golden Rule Exercises

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 $R\sin (x+\alpha )$, where $R>0$ and $0<\alpha <\frac{\pi }{2}$.

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

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