Prove the following identities: `(tantheta+sectheta-1)/(tantheta-sectheta+1)=(1+sintheta)/(costheta)`

Explanation

L.H.S `= frac{tan theta+sec theta-1}{tan theta-sec theta+1}`


Identity:` sec ^{2} theta-tan ^{2} theta=1`


`=frac{tan theta+sec theta-(sec ^{2} theta-tan ^{2} theta)}{tan theta-sec theta+1}`


`=frac{tan theta+sec theta-(sec theta-tan theta)(sec theta+tan theta)}{tan theta-sec theta+1}`


`=frac{(tan theta+sec theta){1-(sec theta-tan theta)}}{tan theta-sec theta+1}`


`=frac{(tan theta+sec theta){1-sec theta+tan theta}}{tan theta-sec theta+1}`


`=tan theta+sec theta`


`=frac{sin theta}{cos theta}+frac{1}{cos theta} `


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