Question

prove that –

(sinA+cosecA)2 + (cosA+secA)2 = 7+tan2A+cot2A​

Answer 1

♣ Qᴜᴇꜱᴛɪᴏɴ :

$\boxed{\sf{Prove\:\left(sinA+cosecA\right)^2\:+\:\left(cosA+secA\right)^2\:=\:7+tan^2A+cot^2A}}$

♣ ᴀɴꜱᴡᴇʀ :

$\mathrm{Manipulating\:left\:side}$

$\left(\sin \left(a\right)+\csc \left(a\right)\right)^2+\left(\cos \left(a\right)+\sec \left(a\right)\right)^2$

$\left(\cos \left(a\right)+\sec \left(a\right)\right)^2+\left(\csc \left(a\right)+\sin \left(a\right)\right)^2=$

$\cos ^{2}(a)+2 \cos (a) \sec (a)+\sec ^{2}(a)+\csc ^{2}(a)+2 \csc (a) \sin (a)+\sin ^{2}(a)$

$=\cos ^2\left(a\right)+\csc ^2\left(a\right)+\sec ^2\left(a\right)+\sin ^2\left(a\right)+2\cos \left(a\right)\sec \left(a\right)+2\csc \left(a\right)\sin \left(a\right)$

$\mathrm{Use\:the\:following\:identity}:\quad \cos ^2\left(x\right)+\sin ^2\left(x\right)=1$

$=1+\csc ^2\left(a\right)+\sec ^2\left(a\right)+2\cos \left(a\right)\sec \left(a\right)+2\csc \left(a\right)\sin \left(a\right)$

$\csc ^{2}(a)+\sec ^{2}(a)+2 \cos (a) \sec (a)+2 \csc (a) \sin (a)=\cot ^{2}(a)+\tan ^{2}(a)+6$

$=1+\cot ^2\left(a\right)+\tan ^2\left(a\right)+6$

$\large\boxed{\sf{=7+\tan ^2\left(a\right)+\cot ^2\left(a\right)}}$

$\mathrm{We\:showed\:that\:the\:two\:sides\:could\:take\:the\:same\:form}$

$\Rightarrow \mathrm{True}$

Answer 2

Step-by-step explanation:

I hope this will help you