Prove the following trigonometric identities-1-tan 2-1-sin-1-sin-1

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Trigonometric Ratios of Multiples of an Angle

Prove the fol...

Question

Prove the following trigonometric identities.

(1 + tan²θ) (1 − sinθ) (1 + sinθ) = 1

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(1 + {tan}^{2} θ) (1 + sin θ) (1 - sin θ) = 1

\frac{1 + \tan^{2} θ}{1 + \cot^{2} θ} = {(\frac{1 - \tan θ}{1 - \cot θ})}^{2} = \tan^{2} θ

\sqrt{\frac{\sec θ - 1}{\sec θ + 1}} + \sqrt{\frac{\sec θ + 1}{\sec θ - 1}} = 2 cosec θ

\sqrt{\frac{1 + \sin θ}{1 - \sin θ}} + \sqrt{\frac{1 - \sin θ}{1 + \sin θ}} = 2 \sec θ

\sqrt{\frac{1 + \cos θ}{1 - \cos θ}} + \sqrt{\frac{1 - \cos θ}{1 + \cos θ}} = 2 cosec θ

\frac{\sec θ - 1}{\sec θ + 1} = {(\frac{\sin θ}{1 + \cos θ})}^{2}

\frac{\sin θ + 1 - \cos θ}{\cos θ - 1 + \sin θ} = \frac{1 + \sin θ}{\cos θ}

(1 + {tan}^{2} θ) (1 - sin θ) (1 + sin θ) =

\frac{\cos θ}{1 - \sin θ} = \frac{1 + \cos θ + \sin θ}{1 + \cos θ - \sin θ}

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