Prove that cos x+sinx /cosx-sinx equal to 1+sin2x/cos2x

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The expression you have provided is true because sin 2x = 2*sin x*cos x and cos 2x = (cos x)^2 - (sin x)^2

Let start with (cos x + sin x)/(cos x - sin x)

multiply the numerator and denominator by (cos x + sin x)

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

=> (cos x + sin x)^2/(cos x)^2 - (sin x)^2

=> [(cos x)^2 + 2*sin x*cos x + (sin x)^2]/cos 2x

=> (1 + sin 2x)/cos 2x

This proves that (cos x + sin x)/(cos x - sin x) = (1 + sin 2x)/cos 2x

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lochana2500 | Student, Undergraduate | (Level 1) Valedictorian

Posted on

L:H:S ≡ (cosx+sinx)/(cosx-sinx)

= (cosx+sinx)(cosx+sinx)/(cosx-sinx)(cosx+sinx)

= (cos²x + 2sinx.cosx + sin²x)/(cos²x-sin²x)

we know that sin²x+cos²x=1, 2sinx.cosx = sin2x and cos²x-sin²x=cos2x

= (1 + sin2x)/cos2x

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