Prove the following identities, where the angles involved are acute angles for which the expressions are defined. $$\dfrac{cos A}{(1+sin A)}+\dfrac{(1+sin A)}{cos A}= 2 sec A$$

Asked by Abhisek | 1 year ago |  44

##### Solution :-

Now, take the L.H.S of the given equation.

L.H.S. =$$\dfrac{ cosA + (1+sin A)}{(1+sin A)cos A}$$

$$\dfrac{ [cos^2A + (1+sin A)^2]}{(1+sin A)cos A}$$

$$\dfrac{ (cos^2A + sin^2A + 1 + 2sin A)}{(1+sin A) cos A}$$

Since cos2A + sin2A = 1, we can write it as

$$1+\dfrac{( 1 + 2sin A)}{(1+sin A) cos A}$$

$$\dfrac{(2+ 2sin A)}{(1+sin A)cos A}$$

$$\dfrac{ 2(1+sin A)}{(1+sin A)cos A}$$

$$\dfrac{2}{cos A}$$ = 2 sec A = R.H.S.

L.H.S. = R.H.S.

$$\dfrac{cos A}{(1+sin A)}+ \dfrac{(1+sin A)}{cos A}$$ = 2 sec A

Hence proved.

Answered by Pragya Singh | 1 year ago

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