Prove the identity # sin theta/ (1-cos theta) - 1/sin theta -= 1/tan theta #?

1 Answer
Steve M
May 7, 2018

We seek to prove that:

# sin theta/ (1-cos theta) - 1/sin theta -= 1/tan theta #

Consider the LHS:

# LHS = sin theta/ (1-cos theta) - 1/sin theta #

# \ \ \ \ \ \ \ \ = ( sin^2 theta - (1-cos theta)) / ((1-cos theta)(sin theta)) #

# \ \ \ \ \ \ \ \ = ( 1-cos^2 theta - 1+cos theta) / ((1-cos theta)(sin theta)) #

# \ \ \ \ \ \ \ \ = ( cos theta- cos^2 theta) / ((1-cos theta)(sin theta)) #

# \ \ \ \ \ \ \ \ = ( cos theta(1- cos theta)) / ((1-cos theta)(sin theta)) #

# \ \ \ \ \ \ \ \ = ( cos theta) / (sin theta) #

# \ \ \ \ \ \ \ \ = cot theta #

# \ \ \ \ \ \ \ \ = 1/tan theta #

# \ \ \ \ \ \ \ \ = LHS \ \ \ \ QED #

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