# How do you verify the identity (2+cscthetasectheta)/(cscthetasectheta)=(sintheta+costheta)^2?

Sep 1, 2016

You will need the following identities to prove this identity.

$\csc \beta = \frac{1}{\sin} \beta$

$\sec \beta = \frac{1}{\cos} \beta$

${\sin}^{2} \beta + {\cos}^{2} \beta = 1$

Simplify the left-hand side and expand the right-hand side.

$\frac{2 + \frac{1}{\sin} \theta \times \frac{1}{\cos} \theta}{\frac{1}{\sin} \theta \times \frac{1}{\cos} \theta} = {\sin}^{2} \theta + 2 \sin \theta \cos \theta + {\cos}^{2} \theta$

$\frac{2 + \frac{1}{\sin \theta \cos \theta}}{\frac{1}{\sin \theta \cos \theta}} = 1 + 2 \sin \theta \cos \theta$

$\frac{\frac{2 \sin \theta \cos \theta + 1}{\sin \theta \cos \theta}}{\frac{1}{\sin \theta \cos \theta}} = 1 + 2 \sin \theta \cos \theta$

$\frac{2 \sin \theta \cos \theta + 1}{\sin \theta \cos \theta} \times \frac{\sin \theta \cos \theta}{1} = 1 + 2 \sin \theta \cos \theta$

$2 \sin \theta \cos \theta + 1 = 1 + 2 \sin \theta \cos \theta$

Identity proved!!

Hopefully this helps!