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Trigonometric Identities

Prove the fol...

Question

Prove the following trigonometric identities:

(i) $\sqrt{\frac{1 + s i n A}{1 - s i n A}} = s e c A + t a n A$

(ii) $\sqrt{\frac{1 - c o s A}{1 + c o s A}} + \sqrt{\frac{1 + c o s A}{1 - c o s A}} = 2 c o s e c A$

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Solution

(i) $\sqrt{\frac{1 + s i n A}{1 - s i n A}} = s e c A + t a n A$

$L H S = \sqrt{\frac{1 + s i n A}{1 - s i n A}}$

Rationalize the numerator and denominator with $\sqrt{1 + s i n A}$

$L H S = \sqrt{\frac{(1 + s i n A) (1 + s i n A)}{(1 - s i n A) (1 + s i n A)}}$

$= \sqrt{\frac{(1 + s i n A)^{2}}{1 - s i n^{2} A}}$

$= \sqrt{\frac{(1 + s i n A)^{2}}{c o s^{2} A}}$

$= \frac{1 + s i n A}{c o s A}$

$= \frac{1}{c o s A} + \frac{s i n A}{c o s A}$

$= s e c A + t a n A$

$= R H S$

(ii) $\sqrt{\frac{1 - c o s A}{1 + c o s A}} + \sqrt{\frac{1 + c o s A}{1 - c o s A}} = 2 c o s e c A$

$L H S = \sqrt{\frac{1 - c o s A}{1 + c o s A}} + \sqrt{\frac{1 + c o s A}{1 - c o s A}}$

Rationalize the numerator and denominator,

$= \sqrt{\frac{(1 - c o s A) (1 - c o s A)}{(1 + c o s A) (1 - c o s A)}} + \sqrt{\frac{(1 + c o s A) (1 + c o s A)}{(1 - c o s A) (1 + c o s A)}}$

$= \sqrt{\frac{(1 - c o s A)^{2}}{1 - c o s^{2} A}} + \sqrt{\frac{(1 + c o s A)^{2}}{1 - c o s^{2} A}}$

$= \sqrt{\frac{(1 - c o s A)^{2}}{s i n^{2} A}} + \sqrt{\frac{(1 + c o s A)^{2}}{s i n^{2} A}}$

$= \frac{(1 - c o s A)}{s i n A} + \frac{(1 + c o s A)}{s i n A}$

$= \frac{1}{s i n A} - \frac{c o s A}{s i n A} + \frac{1}{s i n A} + \frac{c o s A}{s i n A}$

$= c o s e c A - t a n A + c o s e c A + t a n A$

$= 2 c o s e c A$

$= R H S$

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Q. Prove the following trigonometric identities.

(i)

\sqrt{\frac{1 + \sin A}{1 - \sin A}} = \sec A + \tan A

\frac{1 + cos A}{sin A} + \frac{sin A}{1 + cos A} = 2 cosec A

\frac{sin A}{1 + cos A} + \frac{1 + cos A}{sin A} = 2 c o s e c A .

\frac{sin A}{1 + cos A} + \frac{1 + cos A}{sin A} = 2 cosec A

Prove the following trigonometric identities:

$\frac{c o s A}{1 - t a n A} + \frac{s i n A}{1 - c o t A} = s i n A + c o s A$

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