My Math Forum Last problem of the week

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 February 24th, 2013, 08:51 PM #1 Senior Member   Joined: Aug 2012 From: South Carolina Posts: 866 Thanks: 0 Last problem of the week says to verify using reciprocals and pythagorean identities 7cota/csca-1 = 7csca+7/cota this is the last one I need assistance with and thanks so much to Mark,Greg,Soroban and now mathman!!! Thank You so much!
 February 24th, 2013, 09:04 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 522 Math Focus: Calculus/ODEs Re: Last problem of the week Start with the left side. 1.) Multiply by $1=\frac{\sin(a)}{\sin(a)}$ 2.) Multiply by $1=\frac{1+\sin(a)}{1+\sin(a)}$ 3.) Apply Pythagorean identity to denominator. 4.) Reduce, then split into sum and use $\sec(\theta)\equiv\frac{1}{\cos(\theta)}$ and $\cot(\theta)\equiv\frac{1}{\tan(\theta)}$.
February 24th, 2013, 09:20 PM   #3
Math Team

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Re: Last problem of the week

Hello, mathkid!

Quote:
 $\text{Verify: }\:\frac{7\,\!\cot\theta}{\csc\theta\,-\,1} \:=\: \frac{7\,\!\csc\theta\,+\,7}{\cot\theta}$

$\text{Left side: }\:\frac{7\,\!\cot\theta}{\csc\theta\,-\,1}$

$\text{Multiply by }\frac{\csc\theta\,+\,1}{\csc\theta\,+\,1}:\;\;\fr ac{7\,\!\cot\theta}{\csc\theta\,-\,1}\,\cdot\,\frac{\csc\theta\,+\,1}{\csc\theta\,+ \,1} \;=\; \frac{7\,\!\cot\theta(\csc\theta\,+\,1)}{\csc^2\th eta\,-\,1} \;=\; \frac{7\,\!\cot\theta(\csc\theta\,+\,1)}{\cot^2\th eta}$

[color=beige]. . . . . . . . . . . . . . . . [/color]$=\;\frac{7(\csc\theta\,+\,1)}{\cot\theta} \;=\; \frac{7\csc\theta\,+\,7}{\cot\theta}$

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