Question 13(b)

Question 13(b)

(i) Show that $\displaystyle \cos^4{x} + \sin^4{x} = \frac{1+\cos^2{(2x)}}{2}$

(ii) Hence, or otherwise, evaluate $\displaystyle \int^{\frac{\pi}{4}}_0\left(\cos^4{x} + \sin^4{x} \right) dx$

Solution

For part (i):

We apply the double angle identities to show the equivalence. Recall that

$$\begin{aligned} \cos{2x} &= \cos^2{x} - \sin^2{x} \\ &= 1 - 2\sin^2{x} \\ &= 2\cos^2{x} -1 \\ \sin{2x} &= 2\sin{x}\cos{x} \end{aligned}$$

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