<s>[INST] Proving the trig identity $\frac{1-\cos\theta}{\sin\theta} = \frac{\sin\theta}{1+\cos\theta}$ without cross-multiplying I need to prove the following identity.

$$\frac{1-\cos\theta}{\sin\theta} = \frac{\sin\theta}{1+\cos\theta}$$

I want to prove it by deduction rather than cross multiplying.
 [/INST] You could use the half angle formulae .
$\large\frac{1-\cos\theta}{\sin\theta}$
$=\large\frac{2\sin^2\frac\theta2}{2\sin\frac\theta2.cos\frac\theta2}$
$= \large\frac{2\sin\frac\theta2}{2\cos\frac\theta2}$ 
multiply both numerator and denominator by $\cos\frac\theta2$
$=\large\frac{ 2\sin\frac\theta2\cos\frac\theta2}{2\cos^2\frac\theta2}$
$= \large\frac{\sin\theta}{\cos\theta+1}$
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