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5. Prove the following identities, where the angles involved are acute angles for which the expressions are defined. cos A/(1+sin A) + (1+sin A)/cos A = 2 sec A

5. Prove the following identities, where the angles involved are acute angles for which the expressions are defined. 
  cos A/(1+sin A) + (1+sin A)/cos A = 2 sec A

Grade:12

1 Answers

Harshit Singh
askIITians Faculty 5963 Points
3 years ago
Dear Student

(cos A/(1+sin A)) + ((1+sin A)/cos A) = 2 sec A Now, take the L.H.S of the given equation.

L.H.S. = (cos A/(1+sin A)) + ((1+sin A)/cos A)
= [cos2A + (1+sin A)^2]/(1+sin A)cos A
= (cos^2 A + sin^2 A + 1 + 2sin A)/(1+sin A) cos A

Since cos^2 A + sin^2 A = 1, we can write it as



= (1 + 1 + 2sin A)/(1+sin A) cos A

= (2+ 2sin A)/(1+sin A)cos A

= 2(1+sin A)/(1+sin A)cos A

= 2/cos A = 2 sec A = R.H.S.
L.H.S. = R.H.S.

(cos A/(1+sin A)) + ((1+sin A)/cos A) = 2 sec A
Hence proved.

Thanks

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