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# Prove the Following Trigonometric Identities. (Sec a - Tan A)/(Sec a + Tan A) = (Cos^2 A)/(1 + Sin A)^2 - CBSE Class 10 - Mathematics

#### Question

Prove the following trigonometric identities.

(sec A - tan A)/(sec A + tan A) = (cos^2 A)/(1 + sin A)^2

#### Solution

We need to prove  (sec A - tan A)/(sec A + tan A) = (cos^2 A)/(1 + sin A)^2

Here, we will first solve the LHS.

Now using sec theta = 1/cos theta and tan theta = sin theta/cos theta, we get

(sec A - tan A)/(sec A + tan A) = (1/cos A - sin A/cos A)/(1/cos A + sin A/cos A)

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

= (1 - sin A)/(1 + sin A)

Further, multiplying both numerator and denominator by 1 + sin A we get

(1 - sin A)/(1 + sin A) = ((1 - sin A)/(1 + sin A))((1 + sin A)/(1 =  sin A))

= ((1 -sin A)(1 + sin A))/(1 + sin A)^2

= (1 s sin^2 A)/(1 + sin A)^2

Now, using the property cos^2 theta + sin^2 theta = 1, we get

So,

(1 - sin^2 A)/(1 + sin A)^2  = cos^2 A/(1 + sin A)^2  = RHS.

Hence proved

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Solution Prove the Following Trigonometric Identities. (Sec a - Tan A)/(Sec a + Tan A) = (Cos^2 A)/(1 + Sin A)^2 Concept: Trigonometric Identities.
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