Integrate $(\cos x)^4$. I see solutions using power reduction everywhere. I vaguely remember doing it based on some manipulation of trig identities $(\cos x)^2 = 1 - (\sin x)^2$ and $u$-substitution alone.
Anybody know what I am talking about?
Integrate $(\cos x)^4$. I see solutions using power reduction everywhere. I vaguely remember doing it based on some manipulation of trig identities $(\cos x)^2 = 1 - (\sin x)^2$ and $u$-substitution alone.
Anybody know what I am talking about?
Using basic trigonometric identities, we have
$\begin{aligned}\cos^4(x)&=\cos^2(x)(1-\sin^2(x))\\ &=\cos^2(x)-\sin^2(x)\cos^2(x)\\ &=\cos^2(x)-\dfrac{\sin^2(2x)}{4}\\ &=\dfrac{1+\cos(2x)}{2}-\dfrac{1-\cos(4x)}{8},\end{aligned}$
which should be much more manageable.