Question

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

Answer 1

Answer with Step-by-step explanation:

Given :  

tanA/(1 + secA) − tanA/(1 − secA) = 2 cosec A

LHS : (sinA/cosA)/(1 + 1/cosA) − (sinA/cosA)/ (1 - 1/cosA)

[By using an  identity, tanθ = sinθ/cosθ ] [secθ = 1/cosθ]

= (sinA/cosA)/[(cosA + 1)/ cosA] − (sinA/cosA)/ [(cosA − 1) /cosA]

[By taking LCM]

= [(sinA/cosA) × cosA /(cosA + 1)] − [(sinA/cosA) × cosA / (cosA − 1)]

= sinA/(cosA + 1) − sinA/(cosA − 1)

= sinA[(1/cosA+ 1) − (1/ cosA−1)]

[Taking sinA common ]

= sinA(cosA − 1 − cosA − 1) / (cos²A−1)

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

= sinA(−2)/(−sin²A)

[By using  an identity, (1- cos²θ) = sin²θ]

= 2/sinA

= 2 × 1/sinA

= 2 cosec A

[cosecA = 1/sinA]

tanA/(1 + secA) − tanA/(1 − secA) = 2 cosec A

L.H.S = R.H.S  

Hence Proved..

HOPE THIS ANSWER WILL HELP YOU…

Answer 2

Answer:

Hey mate plzz refer to the attachment,

In simple solution,,

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