Question

Prove the following trigonometric identities. $\frac{1}{secA-1}+\frac{1}{secA+1}=2cosecAcotA$

Answer 1

Answer with Step-by-step explanation:

Given :  

1/(secA –1) + 1 / (secA + 1) = 2cosecAcotA

LHS : 1/(secA –1) + 1/(secA + 1)

= (secA + 1 + secA − 1) / [(secA + 1)(secA − 1)]

[By taking LCM]

= 2secA/(sec²A − 1)

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

= 2secA/(tan²A)

[By using  an identity, sec²θ - 1 = tan²θ]

= 2 × 1/cosA / [(cos²A/sin²A)]

= 2/cosA × sin²A/cos²A

= 2(cosA/sin²A)

= 2 × cosA/sinA × 1/sinA

= 2 cotA × cosecA

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

= 2cosec A cot A

1/(secA –1) + 1/(secA +1) = 2cosecAcotA

L.H.S = R.H.S  

Hence Proved..

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Answer 2

Answer:

Refer to the attachment for your answer.