Question

Prove that Sectheta - tantheta +1 / Sectheta + tantheta +1 = 1 - Sintheta/costheta​

Answer 1

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$\displaystyle \: \frac{ (sec \theta - tan \theta + 1 \: )}{( sec \theta +tan \theta + 1\: )}$

$= \displaystyle \: \frac{ (sec \theta - tan \theta + sec²\theta- tan²\theta}{ ( sec \theta +tan \theta + 1\: )}$

$= \displaystyle \: \frac{[(sec \theta - tan \theta) + (sec \theta+tan \theta) (sec \theta - tan \theta)}{ ( sec \theta +tan \theta + 1\: )}$

$= \displaystyle \: \frac{(sec\theta - tan \theta)( sec \theta +tan \theta + 1\: ) }{( sec \theta +tan \theta + 1\: )}$

$\displaystyle \:= (sec \theta - tan \theta \: )$

$\displaystyle \: = \frac{( 1 - sin \theta ) }{cos \theta}$

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Answer 2

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