Prove the following identities- cos-A-cosec-A-sin-A-sec-A-cos-A-sin-A-cosec-A-sec-A

LHS = cos A cosec A sin A sec A/cos A+sin A = cos A cosec A sin A sec A/cos A+sin A = cosA ×1/sinA sinA ×1/cosA/cosA+sinA = cosA/sinA sinA/cosA/cosA+sinA ...

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Standard X

Mathematics

Trigonometric Identities

Prove the fol...

Question

Prove the following identities:

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

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Solution

$L H S = \frac{c o s A c o s e c A - s i n A s e c A}{c o s A + s i n A}$

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

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

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

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

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

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

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

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

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

$= c o s e c A - s e c A = R H S$

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Prove the following identities: cos A cosec A−sin A sec Acos A+sin A=cosec A−sec A

Prove the following identities:

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