Question

Which of the following expressions is equivalent to the expression:

sin A - cos A / sin A - cos A +
sin A + cos A / sin A - cos A

a) 2/2 cos² A-1
b) 2/2 sin² A-1
c) 4 sinA cosA/ 1-2 sin²A
d)4 sinA cosA/ 2cos² A-1​

Answer 1

$\sf \bold{Given\::}$ $\sf \dfrac{sinA- cos A}{sin A + cos A} + \dfrac{sin A + cos A}{sin A - cos A}$

$\sf \bold{To\:Find\::}$ Equivalent expression

$\sf \bold{Identity\:used\::}$

$\sf\bullet\:(a+b)^{2} = a^{2} + b^{2}\:+\:2(a)(b)\\\\\sf \bullet\:(a-b)^{2} = a^{2} + b^{2}\:-\: 2(a)(b)\\\\\sf \bullet sin^{2}x + cos^{2} x\: = \: 1\\\\\sf \bullet\:(a+b)(a-b)\:=\:(a^{2})-(b^{2})$

$\sf \bold{Solution\::}$

$\sf \dfrac{sinA- cos A}{sin A + cos A} + \dfrac{sin A + cos A}{sin A - cos A}\\\\\sf taking\: LCM\:,we\:get;\\\\\sf \implies \dfrac{(\:(sinA- cos A)(sin A - cos A)\:) + (\:(sinA\:+\:cos A)(sin A + cos A)\:) }{ (sinA+ cos A)(sin A - cos A) }\\\\\\\sf \implies \dfrac{\:(sinA- cos A)^{2}+\:(sinA\:+\:cos A)^{2} }{ (sin^{2}A\:-\:cos^{2}A)}\\\\\\\sf \implies \dfrac{\:(sin^{2}A\:+\: cos^{2}A\:-\:2sinAcosA)\:+\:(sin^{2}A\:+\:cos^{2}A\:+\:2sinAcosA) }{ (sin^{2}A\:-\:cos^{2}A)}$

$\sf \implies \dfrac{\:1-\:2sinAcosA\:+\:1\:+\:2sinAcosA }{ (sin^{2}A\:-\:(1-sin^{2}A)}\\\\\\\sf \implies \dfrac{\:1+1 }{ (sin^{2}A\:-\:1\:+\:sin^{2}A)}\\\\\\\sf \implies \dfrac{\:2}{ 2sin^{2}A\:-\:1\:}$

$\sf \bold{ANSWER\::}$ Option $\sf \bold{b)\:\dfrac{2}{2sin^{2}A-1} \:}$