Prove that ( + cosec )^2 + ( + sec )^2 = 7 + ^2 + ^2 . - Vidyadip

Question

Prove that $(\sin \theta + cosec \theta )^2 + (\cos \theta + sec \theta )^2 = 7 + \tan^2\theta + \cot^2\theta .$

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Answer

$L.H.S = (\sin \theta + cosec \theta )^2 + (\cos \theta + sec \theta )^2$
$= (\sin^2\theta + cosec^2\theta + 2 \sin \theta cosec \theta + \cos^2\theta + sec^2\theta + 2cos \theta sec \theta )$
$= (\sin^2\theta + \cos^2\theta ) + (cosec^2\theta + sec^2\theta ) + 2 \sin \theta \left(\frac{1}{\sin \theta}\right)+2 \cos \theta\left(\frac{1}{\cos \theta}\right)$
$= (1) + (1 + \cot^2\theta + 1 + \tan^2\theta ) + (2) + (2)$
$= 7 + \tan^2\theta + \cot^2\theta$
$= R.H.S$
$$

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