# Prove that sec*2 theeta + cosec*2 theeta =sec*2 theeta x cosec*2 theeta?

Mar 1, 2018

Proved in explanation...

#### Explanation:

${\sec}^{2} \theta + \cos e {c}^{2} \theta$ can be written as

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

Taking lcm

$\frac{{\sin}^{2} \theta + {\cos}^{2} \theta}{{\sin}^{2} \theta \cdot {\cos}^{2} \theta}$

$\textcolor{b l u e}{\text{recall} {\sin}^{2} A + C o {s}^{2} A = 1}$

$\frac{1}{{\sin}^{2} \theta \cdot {\cos}^{2} \theta}$

$\frac{1}{\sin} ^ 2 \theta \cdot \frac{1}{\cos} ^ 2 \theta$

cosec^2 theta×sec^2 theta
Hence proved

Mar 1, 2018

See below.

#### Explanation:

${\sec}^{2} \left(\theta\right) + {\csc}^{2} \left(\theta\right) = {\sec}^{2} \left(\theta\right) {\csc}^{2} \left(\theta\right)$

Identities:

1) $\textcolor{red}{\boldsymbol{{\sec}^{2} \left(x\right) = \frac{1}{\cos} ^ 2 \left(x\right)}}$

2) $\textcolor{red}{\boldsymbol{{\csc}^{2} \left(x\right) = \frac{1}{\sin} ^ 2 \left(x\right)}}$

3) $\textcolor{red}{\boldsymbol{{\sin}^{2} x + {\cos}^{2} x = 1}}$

$L H S$

${\sec}^{2} \left(\theta\right) + {\csc}^{2} \left(\theta\right)$

Using identities 1 and 2:

$\frac{1}{\cos} ^ 2 \left(\theta\right) + \frac{1}{\sin} ^ 2 \left(\theta\right)$

$\frac{{\sin}^{2} \left(\theta\right) + {\cos}^{2} \left(\theta\right)}{{\cos}^{2} \left(\theta\right) {\sin}^{2} \left(\theta\right)}$

Using identity 3

$\frac{1}{{\cos}^{2} \left(\theta\right) {\sin}^{2} \left(\theta\right)}$

By identities 1 and 2:

${\sec}^{2} \left(\theta\right) {\csc}^{2} \left(\theta\right)$

$L H S \equiv R H S$