\(\rm \frac{1+tan^2\theta}{1+cot^2\theta}-\left(\frac{1-tan\theta}{1-cot\theta}\right)^2\) किसके बराबर है?

संकल्पना:

त्रिकोणमितीय सूत्र

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 = \(\rm \frac{1}{sin \theta }\)

गणना:

\(\rm \frac{1+tan^2\theta}{1+cot^2\theta}-\left(\frac{1-tan\theta}{1-cot\theta}\right)^2\)

\(= \frac{sec^2θ}{cosec^2θ}-\left(\frac{1-tanθ}{1-\frac{1}{tan θ }}\right)^2\)

\(= \rm \frac{\frac{1}{cos^{2}θ}}{\frac{1}{sin^{2}θ}} - \left (\frac{1 - tanθ}{\frac{tan θ - 1}{tan θ }} \right )^{2}\)

\(\rm = \frac{sin^{2}θ }{cos^{2} θ } - \left (\frac{1 - tanθ}{\frac{- (1 - tan θ )}{tan θ }} \right )^{2}\)

= tan² θ - (-tan θ)²

= tan2 θ - tan2 θ 

= 0