What is the value of tan2 θ−sec2 θcot2 θ−cosec2 θ.
Question:

What is the value of $\frac{\tan ^{2} \theta-\sec ^{2} \theta}{\cot ^{2} \theta-\operatorname{cosec}^{2} \theta}$.

Solution:

We have,

$\frac{\tan ^{2} \theta-\sec ^{2} \theta}{\cot ^{2} \theta-\operatorname{cosec}^{2} \theta}=\frac{-1\left(\sec ^{2} \theta-\tan ^{2} \theta\right)}{-1\left(\operatorname{cosec}^{2} \theta-\cot ^{2} \theta\right)}$

$=\frac{\sec ^{2} \theta-\tan ^{2} \theta}{\operatorname{cosec}^{2} \theta-\cot ^{2} \theta}$

We know that,

$\sec ^{2} \theta-\tan ^{2} \theta=1$

 

$\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1$

Therefore,

$\frac{\tan ^{2} \theta-\sec ^{2} \theta}{\cot ^{2} \theta-\operatorname{cosec}^{2} \theta}=\frac{1}{1}$

$=1$

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