Mathematics: (tan theta+ cot theta)^2= sec^2 theta+cosec^2 theta

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(tan theta+ cot theta)^2= sec^2 theta+cosec^2 theta

naomicervero

12.01.2022

Proved that $(\tan \theta + \cot \theta)^{2} = \sec^{2} \theta + \csc^{2} \theta$ .

Step-by-step explanation:

We have to prove that

$(\tan \theta + \cot \theta)^{2} = \sec^{2} \theta + \csc^{2} \theta$

Now, Left hand side

= $(\tan \theta + \cot \theta)^{2}$

= $\tan^{2} \theta + 2\times \tan \theta \times \cot \theta + \cot^{2} \theta$ {Since we know the formula (a + b)² = a² + 2ab + b²}

= $\tan^{2} \theta + 2 + \cot^{2} \theta$

{Since $\tan \theta \times \cot \theta = 1$ }

= $(\tan^{2} \theta + 1) + (\cot^{2} \theta + 1)$

= $\sec^{2} \theta + \csc^{2} \theta$

{Since we know the identities:

$\sec^{2} x = \tan^{2} x + 1$ and $\csc^{2} x = \cot^{2} x + 1$ }

= Right hand side

Hence, proved.

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