My Math Forum Trigonomic Identity!! Driving me CRAZY

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 January 28th, 2011, 05:41 PM #1 Newbie   Joined: Nov 2010 Posts: 9 Thanks: 0 Trigonomic Identity!! Driving me CRAZY Sin^2?+2cos?-1..........1 --------------------- = ------------ 2+cos?-cos^2?........1+sec? I've been at it for 2 days and I can't friggen get it! Please help DX (The "....." is meant to show the gap between fractions, the forum ignores numerous spaces!"
 January 28th, 2011, 05:51 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,193 Thanks: 504 Math Focus: Calculus/ODEs Re: Trigonomic Identity!! Driving me CRAZY Let's start with the left side: $\frac{\sin^2\theta+2\cos\theta-1}{2+\cos\theta-\cos^2\theta}$ Use the Pythagorean identity in the numerator, i.e $\sin^2\theta=1-\cos^2\theta$ $\frac{1-\cos^2\theta+2\cos\theta-1}{2+\cos\theta-\cos^2\theta}$ Collect like terms: $\frac{-\cos^2\theta+2\cos\theta}{2+\cos\theta-\cos^2\theta}$ Factor numerator and denominator: $\frac{\cos\theta(2-\cos\theta)}{(1+\cos\theta)(2-\cos\theta)}$ Divide out common factors: $\frac{\cos\theta}{\cos\theta+1}$ Divide numerator and denominator by $\cos\theta$ $\frac{1}{1+\sec\theta}$
January 28th, 2011, 06:05 PM   #3
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Re: Trigonomic Identity!! Driving me CRAZY

Quote:
 Originally Posted by MarkFL $\frac{\cos\theta}{\cos\theta+1}$ Divide numerator and denominator by $\cos\theta$ $\frac{1}{1+\sec\theta}$

Can you please elaborate on this step?

Why are we dividing Num and Denom by Cos theta?
And how does that turn it into Sec theta?

 January 28th, 2011, 06:09 PM #4 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,193 Thanks: 504 Math Focus: Calculus/ODEs Re: Trigonomic Identity!! Driving me CRAZY If we do this division, then we get the expression into the required form. $\frac{\cos\theta}{\cos\theta+1}=\frac{\frac{\cos\t heta}{\cos\theta}}{\frac{\cos\theta+1}{\cos\theta} }=\frac{1}{\frac{\cos\theta}{\cos\theta}+\frac{1}{ \cos\theta}}=\frac{1}{1+\sec\theta}$

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