Prove the following trigonometric identities-tan -160- -952-1tan -160- -952-sec -160- -952- -160-cosec -160- -952-

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Trigonometric Ratio(sin,cos,tan)

Prove the fol...

Question

Prove the following trigonometric identities.

$\tan θ + \frac{1}{\tan θ} = \sec θ cosec θ$

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\frac{t a n θ}{s e c θ - 1} + \frac{t a n θ}{s e c θ + 1} = 2 c o s e c θ

\sqrt{\frac{s e c θ - 1}{s e c θ + 1}} + \sqrt{\frac{s e c θ + 1}{s e c θ - 1}} = 2 c o s e c θ

In the table, trigonometric ratios and the intervals of $θ$ are given. Match the intervals with the ratios which are positive in those intervals.

\begin{matrix} θ & gives positive values p. (0, \frac{π}{2}) & 1. Only sin θ, cosec θ q. (\frac{π}{2}, π) & 2. Only cos θ, sec θ r. (π, \frac{3 π}{2}) & 3. Only tan θ, cot θ s. (\frac{3 π}{2}, 2 π) & 4. All sin θ, cos θ, tan θ, cot θ, sec θ, cosec θ \end{matrix}

Prove the following trigonometric identities:

$\frac{t a n θ}{1 - c o t θ} + \frac{c o t θ}{1 - t a n θ} = 1 + t a n θ + c o t θ$

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