Prove that-tan theta - cot theta -2 sin 2 theta -1 - sin theta cos theta -1-2 cos 2 theta - sin theta cos theta

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Trigonometric Identities

Prove that:ta...

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Prove that:

tan theta - cot theta = 2sin²theta - 1/sin theta cos theta = 1-2cos²theta /sin theta cos theta

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Prove the following trigonometric identities.
(i) $\frac{1 + cos θ + sin θ}{1 + cos θ - sin θ} = \frac{1 + sin θ}{cos θ}$

(ii) $\frac{sin θ - cos θ + 1}{sin θ + cos θ - 1} = \frac{1}{sec θ - tan θ}$

(iii) $\frac{cos θ - sin θ + 1}{cos θ + sin θ - 1} = cosec θ + cot θ$

(iv) $(sin θ + cos θ) (tan θ + cot θ) = sec θ + cosec θ$

(1 + cot θ + tan θ) (sin θ - cos θ) = sin θ tan θ - cot θ cos θ

\frac{2 \sin θ \cos θ - \cos θ}{1 - \sin θ + \sin^{2} θ - \cos^{2} θ} = \cot θ

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