Prove the following identity (1+2(sin a)(cos a))/(1-2(sin^2 a)=(cos a+sin a)/(cos a-sin a)

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We have to prove that (1+2(sin a)(cos a))/(1-2(sin^2 a)=(cos a+sin a)/(cos a-sin a)

Starting from the left hand side

(1+2(sin a)(cos a))/(1-2(sin a)^2)

replace 1 with (sin a)^2 + (cos a)^2

=> [(sin a)^2 + (cos a)^2 + 2(sin a)(cos a)] / [(sin a)^2 + (cos a)^2 - 2(sin a)^2]

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We have to prove that (1+2(sin a)(cos a))/(1-2(sin^2 a)=(cos a+sin a)/(cos a-sin a)

Starting from the left hand side

(1+2(sin a)(cos a))/(1-2(sin a)^2)

replace 1 with (sin a)^2 + (cos a)^2

=> [(sin a)^2 + (cos a)^2 + 2(sin a)(cos a)] / [(sin a)^2 + (cos a)^2 - 2(sin a)^2]

=> ( sin a  + cos a)^2 / [(cos a)^2 - (sin a)^2]

=> ( sin a  + cos a)^2 / [(cos a - sin a)(cos a + sin a)]

=> (sin a  + cos a) / ( cos a  - sin a)

which is the right side side

We prove (1+2(sin a)(cos a))/(1-2(sin^2 a)=(cos a+sin a)/(cos a-sin a)

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