# Prove the given identities. 1. cos^4(x)-sin^4(x) = cos(2x) 2. (1-cos(X))/(1+cos(X))=tan^2 (x/2)

Prove the given identities.
1.${\mathrm{cos}}^{4}\left(x\right)-{\mathrm{sin}}^{4}\left(x\right)=\mathrm{cos}\left(2x\right)$
2.$\frac{1-\mathrm{cos}\left(X\right)}{1+\mathrm{cos}\left(X\right)}={\mathrm{tan}}^{2}\left(\frac{x}{2}\right)$
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Cheyanne Charles
1) ${\mathrm{cos}}^{4}x-{\mathrm{sin}}^{4}x=\left({\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x\right)\left({\mathrm{cos}}^{2}x+{\mathrm{sin}}^{2}x\right)$
we have
$={\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x$
$={\mathrm{cos}}^{2}x-\left(1-{\mathrm{cos}}^{2}x\right)$
$=2{\mathrm{cos}}^{2}x-1$
$=\mathrm{cos}\left(2x\right)$
2) We have the trignometric relations
$1+\mathrm{cos}\left(2t\right)=2{\mathrm{cos}}^{2}t$
$1-\mathrm{cos}\left(2t\right)=2{\mathrm{sin}}^{2}t$
Putting 2t = x, we get
$1+\mathrm{cos}\left(x\right)=2{\mathrm{cos}}^{2}\left(x/2\right)\to 1$
$1-\mathrm{cos}\left(x\right)=2{\mathrm{sin}}^{2}\left(x/2\right)\to 2$
$2/1⇒{\mathrm{tan}}^{2}\left(x/2\right)$