Question

(Tan theta + sin theta - 1)(tan theta + sec theta +1)=2sin theta /1-sin theta

Answer 1

(tan theta + sec theta -1)/(tan theta -sec...

(tan theta + sec theta -1)/(tan theta -sec theta +1) = sec theta +tan theta

Numerator = tan theta + sec theta -1

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

= (sin theta + 1)/cos theta -1

denominator = (sin theta -1)/cos theta +1

multiply both by cos theta

numerator = sin theta +1 - cos theta

= 2 sin theta/2 cos theta/2 + 2 sin^ theta/2

denominator = 2 sin theta/2 cos theta/2 - 2 sin ^2 theta/2

divide

number = (cos theta/2 + sin theta/2)/(cos theta/2 - sin theta/2)

= (cos theta/2 + sin theta/2)^2/(cos^2 theta/2- sin ^2 theta/2)

= (cos^2 theta/2+ sin^2 theta/2 + 2 cos theta/2 sin theta/2)/(cos theta)

= (1+ sin theta)/ cos theta

= sec theta+ tan theta

Hence Proved

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