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  4. If cos θ=( cos α- cos β/1- cos α cos β), then one of the values of tan (θ/2) is :

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Q. If $\cos \theta=\frac{\cos \alpha-\cos \beta}{1-\cos \alpha \cos \beta}$, then one of the values of $\tan \frac{\theta}{2}$ is :

Trigonometric Functions

Solution:

$\tan ^{2} \frac{\theta}{2}=\frac{1-\cos \theta}{1+\cos \theta}=\frac{1-\frac{\cos \alpha-\cos \beta}{1-\cos \alpha \cos \beta}}{1+\frac{\cos \alpha-\cos \beta}{1-\cos \alpha \cos \beta}}$
$=\frac{1-\cos \alpha \cos \beta-\cos \alpha+\cos \beta}{1-\cos \alpha \cos \beta+\cos \alpha-\cos \beta}$
$=\frac{(1-\cos \alpha)+\cos \beta(1-\cos \alpha)}{(1+\cos \alpha)-\cos \beta(1+\cos \alpha)}$
$=\frac{(1-\cos \alpha)+(1+\cos \beta)}{(1+\cos \alpha)(1-\cos \beta)}=\tan ^{2} \frac{\alpha}{2} \cot ^{2} \frac{\beta}{2}$
$\therefore \tan \frac{\theta}{2}=\pm \tan \frac{\alpha}{2} \cot \frac{\beta}{2}$