Question

prove that sintheta-costheta+1/sintheta+costheta-1=1/sectheta-tantheta​

Answer 1

hope this helped

Step-by-step explanation:

you just need to divide cos∅

Answer 2

SOLUTION

CORRECT QUESTION

TO PROVE

$\displaystyle \sf{ \frac{ \sin \theta - \cos \theta + 1}{\sin \theta + \cos \theta - 1} = \frac{1}{ \sec \theta \: - \tan \theta} }$

PROOF

LHS

$\displaystyle \sf{ = \frac{ \sin \theta - \cos \theta + 1}{\sin \theta + \cos \theta - 1} }$

Dividing numerator and denominator both by cos θ we get

$\displaystyle \sf{ = \frac{ \tan \theta - 1 + \sec \theta }{\tan \theta + 1 - \sec \theta } }$

$\displaystyle \sf{ = \frac{ \tan \theta - 1 + \sec \theta }{\tan \theta - \sec \theta + 1 } }$

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

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

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

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

$\displaystyle \sf{ = \frac{1 }{\sec \theta - \tan \theta } }$

= RHS

Hence proved

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