Question

Prove that ${(\sin \theta +\csc \theta )}^2+{(\cos \theta +\sec \theta )}^2=7+{\tan }^2\theta +{\cos }^2\theta$

Answer 1

$LHS={(\sin \theta +\csc \theta )}^2+{(\cos \theta +\sec \theta )}^2=\sin {}^2\theta +\csc {}^2\theta +2\sin \theta \csc \theta +\cos {}^2\theta +\sec {}^2\theta +2\cos \theta \sec \theta =1+2+2+\csc {}^2\theta +\sec {}^2\theta =1+2+2+1+\cot {}^2\theta +1+\tan {}^2\theta =7+\tan {}^2\theta +\cot {}^2\theta$

LHS=RHS

Hence proof.