Question

1+cos theta+sin theta/1-cos theta+sin theta =1+sin theta by costheta

Answer 1

Answer:

{1+cos\theta + sin\theta}{1+cos\theta - sin\theta}={1+sin\theta}{cos\theta}

L.H.S.

{1+cos\theta + sin\theta}{1+cos\theta - sin\theta}

={1+cos\theta + sin\theta}{1+cos\theta - sin\theta}\times \frac{1+cos\theta + sin\theta}{1+cos\theta + sin\theta}

={(1+cos\theta +sin\theta)^2}{(1+cos\thta)^2-sin^2\theta}

={(1+cos\theta)^2+sin^2\theta + 2 sin\theta(1+cos\theta)}{1+cos^2\theta + 2cos\theta - sin^2 \theta}

={1+cos^2\theta + 2 cos\theta + sin^2\theta + 2 sin\theta(1+cos\theta)}{1+cos^2\theta + 2cos \theta - 1 + cos^2 \theta}

={1+1+2cos \theta + 2 sin \theta( 1+cos\theta)}{2cos^2\theta + 2 cos\theta} (sin^2\theta + cos^2 \theta = 1)

={2+2cos\theta + 2sin\theta cos\theta + 2 sin \theta}{2cos^2 \theta + 2 cos \theta }

={2(1+cos\theta)+2sin \theta ( 1+ cos \theta)}{2cos \theta (cos \theta + 1)}

={1+sin\theta }{cos \theta }

R.H.S.

Hence, proved.