# $\sin x+\cos x=\frac{5}{4}$, find $\cos(4x)$ without a calculator

It is given that $$\sin x+\cos x=\frac{5}{4}$$. I need to find the exact value of $$\cos(4x)$$. I do know how to find it with a calculator, but do not know how to without.

## 3 Answers

If you square both sides you get $$1+2\sin x\cos x=25/16$$ or $$\sin(2x)=9/16$$ Then $$\cos(4x)=1-2\sin^2(2x)=1-162/256=94/256.$$

$$\sin x+\cos x=\sqrt2\sin\left(x+\frac\pi 4\right)$$ $$\cos 4x=-\cos\left(4\left(x+\frac\pi 4\right)\right).$$ Let $$y=x+\pi/4$$. If we know $$\sin y$$ we can work out $$\cos 2y=1-2\sin^2y$$ and then $$\cos4y=2\cos^22y-1$$.

$$(\sin x+\cos x)^2=1+\sin2x=\frac{25}{16}$$ $$\sin2x=\frac{9}{16}$$ $$\cos4x=1-2(\sin2x)^2=\frac{94}{256}$$