# How do you use the angle sum identity to find the exact value of cos 105?

Litzy Downs 2022-09-30 Answered
How do you use the angle sum identity to find the exact value of $\mathrm{cos}105$?
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$\mathrm{cos}\left(a+b\right)=\mathrm{cos}a\mathrm{cos}b-\mathrm{sin}a\mathrm{sin}b$
Therefore,
$\mathrm{cos}105=\mathrm{cos}\left(60+45\right)$
$=\mathrm{cos}60\mathrm{cos}45-\mathrm{sin}60\mathrm{sin}45$
$=\left(\frac{1}{2}\right)×\left(\frac{\sqrt{2}}{2}\right)-\left(\frac{\sqrt{3}}{2}\right)×\left(\frac{\sqrt{2}}{2}\right)$
$=\frac{1}{4}\left(\sqrt{2}-\sqrt{6}\right)$
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Vrbljanovwu
using the trigonometric identity
$\mathrm{cos}\left(x+y\right)=\mathrm{cos}x\mathrm{cos}y-\mathrm{sin}x\mathrm{sin}y$
$\mathrm{cos}105=\mathrm{cos}\left(60+45\right)$
$=\mathrm{cos}60\mathrm{cos}45-\mathrm{sin}60\mathrm{sin}45$
$=\left(\frac{1}{2}×\frac{\sqrt{2}}{2}\right)-\left(\frac{\sqrt{3}}{2}×\frac{\sqrt{2}}{2}\right)$
$=\frac{\sqrt{2}}{4}-\frac{\sqrt{6}}{4}=\frac{1}{4}\left(\sqrt{2}-\sqrt{6}\right)$