Question

Tan theta + sec theta -1. 1 + sin theta ----------------------------------- = ------------------
Tan theta - sec theta +1. Cos theta

Answer 1

$\huge\underline\bold\red{★Answer★}$

$\frac{tan \theta + sec \theta - 1 }{tan \theta - sec \theta + 1 } \\ = \frac{(tan \theta + sec \theta ) - ( {sec}^{2} \theta - {tan}^{2} \theta) }{tan \theta - sec \theta + 1} \\ = \frac{(sec \theta + tan \theta ) - (sec \theta - tan \theta )(sec \theta + tan \theta )}{tan \theta - sec \theta + 1} \\ = \frac{(sec \theta + tan \theta )(1 - sec \theta + tan \theta )}{tan \theta - sec \theta + 1} \\ = sec \theta + tan \theta \\ = \frac{1}{cos \theta } + \frac{sin \theta }{cos \theta } \\ = \frac{1 + sin \theta }{cos \theta }$