Pre-Cal.-Proving Identities

Tutor: None Selected Time limit: 1 Day

How can I prove this identity? I know the answer is "true," but I can't figure out exactly how to arrive at the answer.  

sinØ/(1-cosØ) - (1+cosØ)/sinØ = 0

Jan 18th, 2015

sinØ / (1 + cosØ) = (1 - cosØ) / sinØ 

I'll use a simpler, step by step approach, the same as when on solves an algebraic equation: 

multiply both sides by sinØ (this gets rid of the denominator on the right): 

sin^2Ø / (1 + cosØ) = (1 - cosØ) 

sin^2Ø simply means "sine squared of phi" 

it is the same as writing (sinØ)^2 

next step, multiply both sides by 1 + cosØ 

this gets rid of the denominator on the left 

sin^2Ø = (1 + cosØ)(1 - cosØ) 

sin^2Ø = 1 - cos^2Ø 

sin^2Ø + cos^2Ø = 1 

which is a well-known identity 

Using descriptions from plane geometry (right-angle triangle) 

(opp/hyp)^2 + (adj/hyp)^2 = 1 

(opposite^2 + adjacent^2)/hypotenuse^2 = 1 

Using Pythagorean theorem: 

hypotenuse^2 = opp.^2 + adj.^2 

we replace the top part 

hypotenuse^2 / hypotenuse^2 = 1 

which is true. 

You have 

sinØ-sinØcosØ/sinØsinØcosØ = cosØ/sinØsinØcosØ. 

I do not see your reasoning. 

At first, I thought you were going for a common denominator, but that would have been (for the denominator) 

(1 + cosØ)sinØ = sinØ + sinØcosØ 

(did you lose the + along the way?) 

Next, you would multiply each side by the missing part of the common denominator (which, in this case, is the denominator from the other side of the = sign) 

sinØsinØ / (sinØ + sinØcosØ) = (1-cosØ)(1+cosØ) / (sinØ + sinØcosØ) 

When you reach this stage, where each side (the whole side) is over the same common denominator, then you can ignore the denominator. 

simple example: let's say k is some fixed number and both sides have it as the denominator 

a/k = b/k 

then, obviously, the only way this can be true is if a=b (you can now ignore the k) 

This idea of dropping the common denominator leaves you with 

(sinØ)^2 = 1 - (cosØ)^2 

which becomes 

sin^2Ø + cos^2Ø = 1

Jan 18th, 2015

Jan 18th, 2015
Jan 18th, 2015
Oct 22nd, 2016
Mark as Final Answer
Unmark as Final Answer
Final Answer

Secure Information

Content will be erased after question is completed.

Final Answer