Question

$\frac{sin2 \theta \: + \: sin \theta}{1 + cos2 \theta + cos \theta} = tan \theta$
prove it ++++

Answer 1

$\: \mathsf {1) \: \frac{sin2 \theta \: + \: sin \theta}{1 \: + \: cos2 \theta \: + \: cos \theta} = tan \theta }$

Solution :-

$\sf{L.H.S} \: = \: \: \mathsf {\frac{sin2 \theta \: + \: sin \theta}{1 \: + \: cos2 \theta \: + \: cos \theta} = tan \theta } \:$

$\mathsf{ \implies \frac{2 \: sin \theta \: cos \theta \: + \: sin \theta}{2 \: {cos}^{2} \theta \: + \: cos \theta } }$

$\mathsf{ \implies \frac{2 \: sin \theta \: ( cos \theta \: + \: 1)}{2 \: cos \theta \: (cos \theta \: + \: 1)} }$

$\mathsf{ \implies \frac{sin \theta}{cos \theta} }$

$\sf { \implies tan \theta \: = \: R.H.S \: proved}$

$\sf{ \therefore \: L.H.S \: = \: R.H.S \: Proved.}$

Answer 2

Answer:

$\dfrac{sin2\theta+sin\theta}{1+cos2\theta+cos\theta}\\\\\implies \dfrac{2sin\theta cos\theta+sin\theta}{2cos^2\theta+cos\theta}\\\\\implies \dfrac{sin\theta(2cos\theta+1)}{cos\theta(2cos\theta+1)}\\\\\implies \dfrac{sin\theta}{cos\theta}\\\\\implies tan\theta$

Step-by-step explanation:

Here we are proving that the above equation is tan theta .

Use sin 2x = 2sinxcosx .

Use cos2x = 2cos²x-1 .

tanx = sinx/cosx .