sin2x / (1+cos2x) = tanx

We will use trigonometric identities to solve.

We will start from the left side and prove the right side.

==> we know that:

sin2x - 2sinx*cosx

cos2x = 2cos^2 x -1

We will substitute.

==> sin2x / (1+ cos2x) = 2sinx*cosx / (1+ 2cos^2 x -1)

= 2sinx*cosx/ 2cos^2 x

We will reduce similar.

==> sin2x / (1+ cos2x) = sinx/cosx

But we know that tanx = sinx/cosx

**==> sin2x / (1+ cos2x) = tanx...........q.e.d**

We have to prove that: sin 2x/ (1 + cos 2x) = tan x

To do this we use the relations : sin 2x = 2 sin x*cos x and cos 2x = (cos x)^2 - (sin x)^2.

sin 2x/ (1 + cos 2x)

=> [2 sin x*cos x] / [1 + (cos x)^2 - (sin x)^2]

=> [2 sin x*cos x] / [1 - (sin x)^2 + (cos x)^2]

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

=> sin x / cos x

=> tan x

**Therefore sin 2x/ (1 + cos 2x) = tan x.**

L:H:S ≡ sin2x/(1 + cos2x)

**⇒ use sin 2A = 2sinA.cosA & cos 2A = 2cos²A - 1**

= 2sinx.cosx/(1+2cos²x - 1)

= 2sinx.cosx/2cos²x

= sinx.cosx/cosx.cosx

= sinx/cosx

= tanx

Therefore L:H:S ≡ R:H:S

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