Prove that; 1-cos(theta)/sin(theta)=1/cosec(theta)+cot(theta)?

1 Answer
Steve M
Jan 14, 2018

We wish to validate the identity:

# (1-cos theta)/(sin theta) -= 1/(csc theta + cot theta) #

Consider the RHS of the expression (as this is more complex that the LHS):

# RHS -= 1/(csc theta + cot theta) #

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

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

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

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

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

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

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

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

Related questions
Impact of this question
4792 views around the world
You can reuse this answer
Creative Commons License