Question

prove that (sin theta+cosec theta)2 + (cos theta +sec theta)2 - (tan theta +cot theta)=5

Answer 1

$right \: \: question : \\ (sin \theta+cosec \theta)^{2} + (cos \theta +sec \theta)^{2} - (tan \theta +cot \theta) \boxed{^{2}}=5 \\ \boxed{^{2} }\: \: is \: \: missing\:\:in \:\:your \:\:question$

Step-by-step explanation:

$LHS=(sin \theta+cosec \theta)^{2} + (cos \theta +sec \theta)^{2} - (tan \theta +cot \theta)^{2} \\ = (sin^{2} \theta + cosec^{2} \theta + 2.sin \theta.cosec \theta) + (cos^{2} \theta + sec^{2} \theta + 2.cos \theta.sec \theta) - (tan^{2} \theta + cot^{2} \theta + 2.tan \theta.cot \theta) \\ = sin^{2} \theta + cosec^{2} \theta + 2 + cos^{2} \theta + sec^{2} \theta + 2- tan^{2} \theta - cot^{2} \theta - 2 \\ = ( sin^{2} \theta + cos^{2} \theta) + (sec^{2} \theta - tan^{2} \theta) + (cosec^{2} \theta - cot^{2} \theta) + (2 + 2 - 2) \\ = \: 1 + 1 + 1 + 2 + 2 - 2 \\ = \: \boxed{ 5}$