Question

Prove the following trigonometric identities. (cosecθ + sinθ) (cosecθ − sinθ) = cot²θ+cos²θ

Answer 1

Answer with Step-by-step explanation:

Given : (cosec θ + sin θ)(cosec θ – sin θ) = cot²θ + cos²θ

L.H.S :  (cosec θ + sin θ)(cosec θ – sin θ)

= cosec²θ – sin² θ

[By using the identity, (a + b)(a – b) = a² – b²]

[By using the identity,cosec²θ =  1 + cot² θ and sin² θ =  1 – cos² θ]

= (1 + cot² θ) – (1 – cos² θ)

= 1 + cot² θ – 1 + cos² θ

= cot² θ + cos²θ

(cosec θ + sin θ)(cosec θ – sin θ) = cot²θ + cos²θ

L.H.S = R.H.S  

Hence Proved..

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Answer 2

Answer:

proved

Step-by-step explanation:

LHS = (cosecθ)^2 - (sinθ)^2

RHS = cot²θ+cos²θ

= cos²θ[ (1/sinθ)^2 + 1]

= [ (sinθ)^2 + 1 ]

----------------------- * [1- (sinθ)^2 ]

(sinθ)^2

= 1- (sinθ)^4

-----------------

(sinθ)^2

= (cosecθ)^2 - (sinθ)^2