# Thread: Half angle or double angle Identities

1. ## Half angle or double angle Identities

I don't understand them at all.
The problem is to evaluate tan(22.5) using exact values.
How should I do this?

2. $\displaystyle \tan \frac{x}{2}=\frac{\sin \frac{x}{2}}{\cos \frac{x}{2}}=\frac{\sqrt{\frac{1-\cos x}{2}}}{\sqrt{\frac{1+\cos x}{2}}}=\sqrt{\frac{1-\cos x}{1+\cos x}}.$

Actually is $\displaystyle \sin \frac{x}{2}=\pm \sqrt{\frac{1-\cos x}{2}},$ but the angle is positive, hence our answer is positive, and that's preserves the positive sign.

3. Originally Posted by Krizalid
$\displaystyle \tan \frac{x}{2}=\frac{\sin \frac{x}{2}}{\cos \frac{x}{2}}=\frac{\sqrt{\frac{1-\cos x}{2}}}{\sqrt{\frac{1+\cos x}{2}}}=\sqrt{\frac{1-\cos x}{1+\cos x}}.$

Actually is $\displaystyle \sin \frac{x}{2}=\pm \sqrt{\frac{1-\cos x}{2}},$ but the angle is positive, hence our answer is positive, and that's preserves the positive sign.
hmm can I assume tan (22.5) is equal to tan 1/2 x?

4. Krizalid is suggesting $\displaystyle x = \frac{\pi}{4}$

5. Well, that was easier than I expected. Thanks.