Question:
Tan a/(sec a - 1) - sin a/(1 + cos a) = 2cot a
Answer:
We have to prove that:
tan a/(sec a - 1) - sin a/(1 + cos a) = 2cot a
LHS:
tan a/(sec a - 1) - sin a/(1 + cos a)
= (sin a/cos a)/(1/cos a - 1) - sin a/(1 + cos a)
= (sin a/cos a)/{(1 - cos a)/cos a} - sin a/(1 + cos a)
= sin a/(1 - cos a) - sin a/(1 + cos a)
= {sin a(1 + cos a) - sin a(1 - cos a)}/{(1 - cos a)*(1 + cos a)}
= {sin a + sin a * cos a - sin a + sin a * cos a)}/(1 - cos2 a)
= {2 * sin a * cos a}/sin2 a
= 2cos a/sin a
= 2cot a
= RHS
So, tan a/(sec a - 1) - sin a/(1 + cos a) = 2cot a
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