Description
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
Explanation & Answer
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
Review
Review
24/7 Homework Help
Stuck on a homework question? Our verified tutors can answer all questions, from basic math to advanced rocket science!
Similar Content
Related Tags
Catching Fire
by Suzanne Collins
To Kill a Mockingbird
by Harper Lee
The Restless Wave
by John McCain
The Knife of Never Letting Go
by Patrick Ness
A Brief History of Humankind Sapiens
by Yuval Noah Harari
The Curious Case of the Dog in the Night Time
by Mark Haddon
Too Much and Never Enough
by Mary L. Trump
The Metamorphosis
by Franz Kafka
Principles - Life and Work
by Ray Dalio