Question

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

Answer 1

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=pi/4`.

`"For "x=pi/4, "The L.H.S."=sec^2(pi/4)csc(pi/4)`

`=(sqrt2)^2sqrt2=2sqrt2`, whereas,

`"The R.H.S.=2+2=4"`.

`:."The L.H.S."!="The R.H.S."`

In fact, `sec^2xcsc^2x=sec^2x+csc^2x`.

`sec^2x+csc^2x=1/cos^2x+1/sin^2x`,

`=((sin^2x+cos^2x))/(cos^2xsin^2x)`,

`=(1)-:{1/sec^2x*1/csc^2x}`,

`=sec^2xcsc^2x`.

Answer 2

`sec^2xcscx!=sec^2x+csc^2x,...why ?`
Please see below.

Explanation:

We know that,

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

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

Here,

`LHS=color(violet)(sec^2xcscx) and RHS=sec^2x+csc^2x`

We take ,

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

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

`=color(red)((sin^2x+cos^2x))/(cos^2xsin^2x)`

`=color(red)(1)/(cos^2xsin^2x)...toApply(2)`

`=1/cos^2x*1/sin^2x`

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

`!=LHS`

Hence, `sec^2xcolor(red)(cscx!=)sec^2x+csc^2x`,

But, `sec^2xcolor(red)(csc^2x=)sec^2x+csc^2x`