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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.

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3 Answers 3

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

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

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

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