# Thread: trig identities

1. ## trig identities

how do i proof this:
((2cos(a)-cos^2(a))/(sin^2(a)+cos(a)+1))=1/(1+sec(a))

2. ## Re: trig identities

Originally Posted by ahmedb
how do i proof this:
((2cos(a)-cos^2(a))/(sin^2(a)+cos(a)+1))=1/(1+sec(a))
\displaystyle \displaystyle \begin{align*} \frac{2\cos{a} - \cos^2{a}}{\sin^2{a} + \cos{a} + 1} &\equiv \frac{\cos{a}\left(2 - \cos{a}\right)}{1 - \cos^2{a} + \cos{a} + 1} \\ &\equiv \frac{\cos{a}\left(2 - \cos{a}\right)}{2 + \cos{a} - \cos^2{a}} \\ &\equiv \frac{\cos{a}\left(2 - \cos{a}\right)}{2 + 2\cos{a} - \cos{a} - \cos^2{a}} \\ &\equiv \frac{\cos{a}\left(2 - \cos{a}\right)}{2\left(1 + \cos{a}\right) - \cos{a}\left(1 + \cos{a}\right)} \\ &\equiv \frac{\cos{a}\left(2 - \cos{a}\right)}{\left(1 + \cos{a}\right)\left(2 - \cos{a}\right)} \\ &\equiv \frac{\cos{a}}{1 + \cos{a}} \\ &\equiv \frac{\cos{a} \left( \frac{1}{\cos{a}} \right) }{ \left( 1 + \cos{a} \right) \left( \frac{1}{\cos{a}} \right) } \\ &\equiv \frac{1}{\frac{1}{\cos{a}} + 1} \\ &\equiv \frac{1}{1 + \sec{a}} \end{align*}

3. ## Re: trig identities

is there a program where you got this from?
or is it your brain?

4. ## Re: trig identities

Originally Posted by ahmedb
is there a program where you got this from?
or is it your brain?
My brain, which is better than any computer program