Question

cosec a- sin a / cosec a + sin a = sec^2 a - tan^2 a /sec^2 a + tan^2 a. ​

Answer 1

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Answer 2

It is true that

cosec a- sin a / cosec a + sin a = sec²a - tan²a /sec²a + tan²a

Given:

cosec a- sin a / cosec a + sin a = sec²a - tan²a /sec²a + tan²a

To find:

Prove that given expression true

Solution:

Given cosec a- sin a / cosec a + sin a = sec²a - tan²a /sec²a + tan²a

Take LHS of given expression

LHS =  cosec a- sin a / cosec a + sin a

As we know cosec = 1/sin

⇒ $\frac{\frac{1}{sin a} - sin a }{ \frac{1}{sin} + sin a }$

⇒ $\frac{\frac{1 - sin^{2}a }{sin a}}{ \frac{1 + sin^{2} a}{sin} }$

⇒ $\frac{ 1 - sin^{2}a}{1 + sin^{2} a }$

⇒ cosec a- sin a / cosec a + sin a = (1 - sin²a)/(1 + sin²a) -----(1)  

RHS = sec²a - tan²a /sec²a + tan²a

sec = 1/cos and tan = sin/cos

⇒  $\frac{\frac{1}{cos^{2}a } - \frac{sin^{2}a }{cos^{2}a } }{\frac{1}{cos^{2}a } + \frac{sin^{2}a }{cos^{2} a} }$

⇒  $\frac{\frac{1 - sin^{2} a}{cos^{2}a } }{\frac{1 +sin^{2}a }{cos^{2}a } }$  

⇒  $\frac{1 - sin^{2} a}{1 +sin^{2}a }$  

⇒ sec²a - tan²a /sec²a + tan²a = (1 - sin²a)/(1 + sin²a) ------(2)

From (1) and (2)

It is proven that

cosec a- sin a / cosec a + sin a = sec²a - tan²a /sec²a + tan²a

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