# 2 Special identities questions..

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• Nov 10th 2008, 10:54 PM
zuuberbat
2 Special identities questions..
a) Use the half-angle formulas for sine and cosine to verify the identity:
tan(a/2) = (1-cosa)/(sina)

b) Let x represent a real number. Find all solutions of the trigonometric equation:
tan(x/2) + sinx = 0

Explanations, steps would be helpful. Thank you!
• Nov 11th 2008, 01:37 AM
mr fantastic
Quote:

Originally Posted by zuuberbat
a) Use the half-angle formulas for sine and cosine to verify the identity:
tan(a/2) = (1-cosa)/(sina)

b) Let x represent a real number. Find all solutions of the trigonometric equation:
tan(x/2) + sinx = 0

Explanations, steps would be helpful. Thank you!

a) Use the double angle formulae:

Substitute $\cos a = \cos \left( 2 \cdot \frac{a}{2} \right) = 1 - 2\sin^2 \left(\frac{a}{2}\right)$ and $\sin a = \sin \left( 2 \cdot \frac{a}{2} \right) = 2 \sin \left(\frac{a}{2}\right) \cos \left(\frac{a}{2}\right)$ into the right hand side and simplify.

b) From a) the equation becomes

$\frac{1 - \cos x}{\sin x} + \sin x = 0$

$\Rightarrow 1 - \cos x + \sin^2 x = 0$

$\Rightarrow 1 - \cos x + (1 - \cos^2 x) = 0$

$\cos^2 x + \cos x - 2 = 0$

$\Rightarrow (\cos x + 2)(\cos x - 1) = 0$

The rest is left for you to do.

NB: There's one small technical difficulty (left for you to spot) which you will need to justify.