Problems:
Trigonometric Identities
The assignment given is to prove that sin θ/(1 - cos θ)] - [(1 + cos θ)/sin θ] = 0. In other words, we want to prove that [sin θ/(1 - cos θ)] = [(1 + cos θ)/sin θ].
sinθ/1-cos θ - 1+cosθ/sinθ = sin2θ-(1 - cosθ)(1+cosθ)/sin θ(1-cosθ) = sin2θ-(1-cos2θ)/sinθ(1-cosθ) = 1-1/sinθ(1-cosθ) = 0.
Here we have used the following identities/operations:
• (a-b)(a+b) = a2 -b2
• sin2 x+cos2 x = 1
• Lowest Common denominator operation.
This problem relates to a combination of arithmetic and trigonometric problems. Whenever there's combination of fraction, one could think of using the lowest common denominator to combine the fraction. Then, you'd think of using trigonometric identities to simplify the equation.
One of the difficulties encountered is to decide which function to use in order to that LHS becomes similar to RHS. My knowledge of trigonometric identities and algebra came to the rescue and helped in simplifying what seemed as a hard function and the solution became easy from that point on.