# What is (costheta)/3 in terms of tantheta?

Dec 28, 2015

costheta/3=+-1/(3sqrt(tan^2theta+1)

#### Explanation:

We know that $S \int h \frac{\eta}{C} o s \theta = T a n \theta$
$\implies S {\in}^{2} \frac{\theta}{\cos} ^ 2 \theta = {\tan}^{2} \theta$
$\implies {\cos}^{2} \theta = {\sin}^{2} \frac{\theta}{\tan} ^ 2 \theta$
Also $S {\in}^{2} \theta = 1 - {\cos}^{2} \theta$

$\implies {\cos}^{2} \theta = \frac{1 - {\cos}^{2} \theta}{\tan} ^ 2 \theta$

$\implies {\cos}^{2} \theta = \frac{1}{\tan} ^ 2 \theta - {\cos}^{2} \frac{\theta}{\tan} ^ 2 \theta$

$\implies {\cos}^{2} \theta + {\cos}^{2} \frac{\theta}{\tan} ^ 2 \theta = \frac{1}{\tan} ^ 2 \theta$

$\implies {\cos}^{2} \theta \left(1 + \frac{1}{\tan} ^ 2 \theta\right) = \frac{1}{\tan} ^ 2 \theta$

$\implies {\cos}^{2} \theta \left(1 + \frac{1}{\tan} ^ 2 \theta\right) = \frac{1}{\tan} ^ 2 \theta$

$\implies {\cos}^{2} \theta = \frac{\frac{1}{\tan} ^ 2 \theta}{1 + \frac{1}{\tan} ^ 2 \theta}$

$\implies {\cos}^{2} \theta = \frac{1}{{\tan}^{2} \theta + 1}$

$\implies \cos \theta = \pm \sqrt{\frac{1}{{\tan}^{2} \theta + 1}}$

$\implies \cos \theta = \pm \frac{1}{\sqrt{{\tan}^{2} \theta + 1}}$

$\implies \cos \frac{\theta}{3} = \pm \frac{\frac{1}{\sqrt{{\tan}^{2} \theta + 1}}}{3}$

implies costheta/3=+-1/(3sqrt(tan^2theta+1)