Correct Answer - Option 1 : 0
Concept:
Trigonometry Formula
sec2 θ = 1 + tan2 θ
cosec2 θ = 1 + cot2 θ
cot θ = \(\rm \frac{1}{tan \theta }\)
sec θ = \(\rm \frac{1}{cos \theta }\)
cosec θ = \(\rm \frac{1}{sin \theta }\)
Calculation:
\(\rm \frac{1+tan^2\theta}{1+cot^2\theta}-\left(\frac{1-tan\theta}{1-cot\theta}\right)^2\)
\(= \frac{sec^2θ}{cosec^2θ}-\left(\frac{1-tanθ}{1-\frac{1}{tan θ }}\right)^2\)
\(= \rm \frac{\frac{1}{cos^{2}θ}}{\frac{1}{sin^{2}θ}} - \left (\frac{1 - tanθ}{\frac{tan θ - 1}{tan θ }} \right )^{2}\)
\(\rm = \frac{sin^{2}θ }{cos^{2} θ } - \left (\frac{1 - tanθ}{\frac{- (1 - tan θ )}{tan θ }} \right )^{2}\)
= tan2 θ - (-tan θ)2
= tan2 θ - tan2 θ
= 0
