Prove the following trigonometric identities- 1-cos-A-1-cos-A-cot-A-cosec-A2

LHS =1 cosA/1+cosA Rationalize the numerator and denominator with (1 – cos A) =(1 cosA)(1 cosA)/(1+cosA)(1 cosA) =(1 cosA)^2/1 cos^2A =(1 cosA)^2/sin^2A =(1 cos ...

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Standard X

Mathematics

Relation between Trigonometric Ratios

Prove the fol...

Question

Prove the following trigonometric identities:

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

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Solution

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

Rationalize the numerator and denominator with (1 – cos A)

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

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

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

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

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

$= (c o s e c A - c o t A)^{2}$

$= (c o t A - c o s e c A)^{2}$

$= R H S$

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Prove the following trigonometric identities: 1−cos A1+cos A=(cot A−cosec A)2

LHS=1−cosA1+cosA Rationalize the numerator and denominator with (1 – cos A) =(1−cosA)(1−cosA)(1+cosA)(1−cosA) =(1−cosA)21−cos2A =(1−cosA)2sin2A =(1−cosAsinA)2 =(1sinA−cosAsinA)2 =(cosecA−cotA)2 =(cotA−cosecA)2 =RHS

Prove the following trigonometric identities:

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