Proving Trigonometric Identity

**itsMoeyy** · April 25th 2009, 03:22 AM

Hey guys can you prove this identity for me and please tell me how you do it. Since its my first time proving identites, I have no idea what to do.

Sin theta + Sin 2 theta
________________________ = Tan theta

1 + Cos theta + Cos 2 theta

Thanks

**ADARSH** · April 25th 2009, 03:36 AM

Originally Posted by itsMoeyy

Hey guys can you prove this identity for me and please tell me how you do it. Since its my first time proving identites, I have no idea what to do.

Sin theta + Sin 2 theta
________________________ = Tan theta

1 + Cos theta + Cos 2 theta

Thanks

$\color{blue}sin(2\theta) = 2sin(\theta)cos(\theta)$

$Numerator= sin(\theta)+2sin(\theta)cos(\theta) = sin(\theta)(1+2cos(\theta))$
-------------------------------------------

$\color{blue}cos(2\theta) = cos^2(\theta)-sin^2(\theta)$

$\color{blue}sin^2(\theta)+cos^2(\theta)=1$

so $cos(2\theta) = 2cos^2(\theta) -1$
--------------------------------------

$Denominator ={\color{red} 1}+cos(\theta) + 2cos^2(\theta) {\color{red}-1} = cos(\theta)(1+2cos(\theta)) <br />$

Fraction now becomes

$L.H.S=\frac{sin(\theta){\color{red}(1+2cos(\theta) )}}{cos(\theta){\color{red}(1+2cos(\theta))}}$

$=\frac{sin(\theta)}{cos(\theta)}$

$=tan(\theta)=R.H.S<br />$

___________________
things in blue are formulas
things in red get canceled

**itsMoeyy** · April 25th 2009, 04:15 AM

Where are the formula's from?

**mr fantastic** · April 25th 2009, 05:01 AM

Originally Posted by itsMoeyy

Where are the formula's from?

They are standard formulae called the double angle formulae. Look for them in your class notes or textbook.

**itsMoeyy** · April 25th 2009, 07:55 AM

Ok, but how do you put the formula's into place? Like do you subtitute them into the equation. But how?

**mr fantastic** · April 25th 2009, 04:27 PM

Originally Posted by itsMoeyy

Ok, but how do you put the formula's into place? Like do you subtitute them into the equation. But how?

You substitute one expression for the other. eg. Replace $\sin (2 \theta)$ with $2 \sin \theta \cos \theta$ etc.

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