Question

Dum hai toh yeh karo.
sec²x+sec²x(1-sec²x)-cosec²x+cosec²x(cosec²x-)
=
cot⁴x - tan⁴x(PROVE)

Answer 1

$\underline{\underline{\Huge{\textbf{Question:}}}}$
$\sf \small{Prove \: that} \\ \sf \small{sec^{2}\, x+ sec^{2}\, x(1-sec^{2}\, x)-cosec^{2}\, x+ cosec^{2}\, x(cosec^{2}\, x-1)} \\ = \sf \small{cot^{4}\, x - tan^{4}\, x}$



$\underline{\underline{\Huge{\textbf{Solution:}}}}$

$\underline{\Large{\textsf{Step-1:}}}$
$\sf\textsf{Consider LHS}$

$\underline{\Large{\textbf{LHS =}}}$

$\small \sf sec^{2}\, x+ sec^{2}\, x(1-sec^{2}\, x)-cosec^{2}\, x+ cosec^{2}\, x(cosec^{2}\, x-1)$

$\underline{\Large{\textsf{Step-2:}}}$
$\sf\textsf{Rewrite terms in LHS using}\\ \sf\textbf{Distributive property of multiplication} \sf\textsf{ and Rearrange.}$


$\sf\textsf{Distributive property of multiplication}$
$\bigstar\: \boxed{\Large{\mathbf{a(b \pm c)\: =\: ab\: \pm ac}}}$

$\small \sf sec^{2}\, x+ sec^{2}\, x-sec^{4}\, x-cosec^{2}\, x+cosec^{4}\, x-cosec^{2}\, x$

$\small \sf 2sec^{2}\, x - sec^{4}\, x - 2cosec^{2}\, x+cosec^{4}\, x$

$\small \sf 2sec^{2}\, x - \left(sec^{2}\, x\right)^{2} - 2cosec^{2}\, x+\left(cosec^{2}\, x\right)^{2}$ –––[1]



$\underline{\Large{\textsf{Step-3:}}}$
$\sf\textsf{Express sec ^{2} \, x in terms of tan ^{2} \, x }\\\textsf{and cosec ^{2} \, x in terms of cot ^{2} \, x}$
$\sf\textsf{(As RHS contains tan and cot)}$

$\textbf{Using the Identities}$
$\bigstar\: \: \boxed{\Large{\mathbf{sec^{2}\, x - tan^{2}\, x = 1}}} \\ \boxed{\Large{\mathbf{cosec^{2}\, x - cot^{2}\, x = 1}}}$

$\implies \bf sec^{2}\, x = 1+tan^{2}\, x$
$\implies \bf cosec^{2}\, x = 1+cot^{2}\, x$

$\sf\textsf{Substitute in [1]}$

$\small \sf 2\left(1+tan^{2}\, x\right)- \left(1+tan^{2}\, x\right)^{2} - 2\left(1+cot^{2}\, x \right)+\left(1+cot^{2}\, x\right)^{2}$ –––[2]



$\underline{\Large{\textsf{Step-4:}}}$
$\sf\textsf{Simplify Equation [2]}$

$\sf\textsf{Using the Identity}$
$\bigstar\: \boxed{\Large{\mathbf{(a+b)^{2}\: = \: a^{2} + b^{2} - 2ab}}}$

$\implies \sf (1+tan^{2}\, x)^{2} = 1 + tan^{4}\, x - 2tan^{2}\, x$
$\implies \sf (1+cot^{2}\, x)^{2} = 1 + cot^{4}\, x - 2cot^{2}\, x$

$\sf\textsf{Substitute in [2]}$

$\small \sf 2\left(1+tan^{2}\, x\right)- \left(1+tan^{4}\, x - 2tan^{2}\, x\right) - 2\left(1+cot^{2}\, x \right)+\left(1+cot^{4}\, x-2cot^{2}\, x\right)$

$\small \sf 2+2tan^{2}\, x - \left(1+tan^{4}\, x - 2tan^{2}\, x\right) - 2 - 2cot^{2}\, x + \left(1+cot^{4}\, x - 2cot^{2}\, x\right)$

$\small \sf \cancel{2}+\cancel{2tan^{2}\, x} - \cancel{1}-tan^{4}\, x + \cancel{2tan^{2}\, x} - \cancel{2} - \cancel{2cot^{2}\, x} + \cancel{1}+cot^{4}\, x - \cancel{2cot^{2}\, x}$

$\sf cot^{4}\, x - tan^{4}\, x$

$\underline{\Large{\textbf{= RHS}}}$

$\underline{\Large{\textsf{Hence Proved}}}$