# How do you prove  sec^2xcscx =sec^2x+csc^2x ?

Apr 21, 2018

Check the Problem.

#### Explanation:

I have doubt about the validity of the ("so called") Identity.

Had it been an Identity, it would have hold good for $x = \frac{\pi}{4}$.

$\text{For "x=pi/4, "The L.H.S.} = {\sec}^{2} \left(\frac{\pi}{4}\right) \csc \left(\frac{\pi}{4}\right)$

$= {\left(\sqrt{2}\right)}^{2} \sqrt{2} = 2 \sqrt{2}$, whereas,

$\text{The R.H.S.=2+2=4}$.

$\therefore \text{The L.H.S."!="The R.H.S.}$

In fact, ${\sec}^{2} x {\csc}^{2} x = {\sec}^{2} x + {\csc}^{2} x$.

${\sec}^{2} x + {\csc}^{2} x = \frac{1}{\cos} ^ 2 x + \frac{1}{\sin} ^ 2 x$,

$= \frac{\left({\sin}^{2} x + {\cos}^{2} x\right)}{{\cos}^{2} x {\sin}^{2} x}$,

$= \left(1\right) \div \left\{\frac{1}{\sec} ^ 2 x \cdot \frac{1}{\csc} ^ 2 x\right\}$,

$= {\sec}^{2} x {\csc}^{2} x$.

Apr 21, 2018

sec^2xcscx!=sec^2x+csc^2x,...why ?

#### Explanation:

We know that,

color(blue)((1)sectheta=1/costheta and csctheta=1/sintheta

color(red)((2)sin^2theta+cos^2theta=1

Here,

$L H S = \textcolor{v i o \le t}{{\sec}^{2} x \csc x} \mathmr{and} R H S = {\sec}^{2} x + {\csc}^{2} x$

We take ,

RHS=color(blue)(sec^2x+csc^2x

=color(blue)(1/cos^2x+1/sin^2x...toApply(1)

$= \frac{\textcolor{red}{\left({\sin}^{2} x + {\cos}^{2} x\right)}}{{\cos}^{2} x {\sin}^{2} x}$

$= \frac{\textcolor{red}{1}}{{\cos}^{2} x {\sin}^{2} x} \ldots \to A p p l y \left(2\right)$

$= \frac{1}{\cos} ^ 2 x \cdot \frac{1}{\sin} ^ 2 x$

=color(violet)(sec^2xcsc^2x

$\ne L H S$

Hence, ${\sec}^{2} x \textcolor{red}{\csc x \ne} {\sec}^{2} x + {\csc}^{2} x$,

But, ${\sec}^{2} x \textcolor{red}{{\csc}^{2} x =} {\sec}^{2} x + {\csc}^{2} x$