Question

Prove the following trigonometric identities. (1 + cot A − cosec A) (1 + tan A + sec A) = 2

Answer 1

Answer with Step-by-step explanation:

Given :   (1 + cot A − cosec A) (1 + tan A + sec A) = 2

LHS : (1 + cot A − cosec A) (1 + tan A + sec A)

= (1 + cos A/sinA - 1/sinA) (1 + sinA/cosA + 1/cosA)

[By using the identity, cotθ = cosθ/sinθ  , tanθ = sinθ/cosθ ] & [ cosecθ = 1/sinθ, secθ = 1/cosθ]

= {(sinA + cos A - 1)/sinA} {(cosA + sinA  + 1)/cosA)}

[By taking LCM]

= {(sinA + cos A - 1) {(cosA + sinA  + 1)}/ sinAcosA

 = {(sinA + cos A)²  - 1²) }/ sinA cosA

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

 

= {(sin²A + cos ²A + 2sinAcosA)  - 1)} / sinA cosA

[By using identity , (a + b)² = a² + 2ab + b²]

= {(sin²A + cos ²A) + 2sinAcosA - 1} / sinA cosA

= 1  + 2sinAcosA - 1 / sinA cosA

[By using the identity , sin² θ + cos² θ = 1]

= 2sinAcosA  / sinA cosA

= 2

(1 + cot A − cosec A) (1 + tan A + sec A) = 2

L.H.S = R.H.S  

Hence Proved..

HOPE THIS ANSWER WILL HELP YOU…

Answer 2

Answer:

It is Proved !!!

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Step-by-step explanation:

Given :

$\: \ \ \ \ \bullet \: (1 + \cot A - \cosec A) (1 + \tan A + \sec A) = 2$

To Find :

• Prove the trigonometric Identity.

Solution :

Given in Attachment.

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Thanks ..!!!