Today i am solving this trigonometry question of ncert of class 10 please help me to find out the best way to solve the exercise 8.4 of trigonometry question no. 5(2) .Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (ii) cos A/(1+sin A) + (1+sin A)/cos A = 2 sec A .
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Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (ii) cos A/(1+sin A) + (1+sin A)/cos A = 2 sec A .Q.5(2)
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(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
= (cos2A + sin2A + 1 + 2sin A)/(1+sin A) cos A
Since cos2A + sin2A = 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.