# trigonometry Prove indentity

• Mar 19th 2009, 08:54 PM
manutd4life
trigonometry Prove indentity
[IMG]file:///C:/DOCUME%7E1/Asus/LOCALS%7E1/Temp/moz-screenshot.jpg[/IMG]Cotx - tanx = 2cot2x

• Mar 20th 2009, 12:04 AM
Prove It
Quote:

Originally Posted by manutd4life
[IMG]file:///C:/DOCUME%7E1/Asus/LOCALS%7E1/Temp/moz-screenshot.jpg[/IMG]Cotx - tanx = 2cot2x

You need to know that

$\displaystyle \cos^2{x} - \sin^2{x} = \cos{2x}$

and

$\displaystyle 2\sin{x}\cos{x} = \sin{2x} \implies \sin{x}\cos{x} = \frac{1}{2}\sin{2x}$.

These are proved using the sum formulas (which are proven here)

Proof of the sum and difference formulas.

Trigonometric identities

$\displaystyle \cot{x} - \tan{x} = \frac{\cos{x}}{\sin{x}} - \frac{\sin{x}}{\cos{x}}$

$\displaystyle = \frac{\cos{x}}{\sin{x}}\times\frac{\cos{x}}{\cos{x }} - \frac{\sin{x}}{\cos{x}}\times\frac{\sin{x}}{\sin{x }}$

$\displaystyle = \frac{\cos^2{x}}{\sin{x}\cos{x}} - \frac{\sin^2{x}}{\sin{x}\cos{x}}$

$\displaystyle = \frac{\cos^2{x} - \sin^2{x}}{\frac{1}{2}\sin{2x}}$

$\displaystyle = \frac{2\cos{2x}}{\sin{2x}}$

$\displaystyle = 2\cot{2x}$.
• Mar 20th 2009, 12:47 AM
manutd4life
i have not understand this part clearly
Quote:

Originally Posted by Prove It

$\displaystyle = \frac{\cos{x}}{\sin{x}}\times\frac{\cos{x}}{\cos{x }} - \frac{\sin{x}}{\cos{x}}\times\frac{\sin{x}}{\sin{x }}$

$\displaystyle = \frac{\cos^2{x}}{\sin{x}\cos{x}} - \frac{\sin^2{x}}{\sin{x}\cos{x}}$

$\displaystyle = \frac{\cos^2{x} - \sin^2{x}}{\frac{1}{2}\sin{2x}}$

please help how u got this $\displaystyle {\frac{1}{2}\sin{2x}}$
• Mar 20th 2009, 01:00 AM
Prove It
Quote:

Originally Posted by manutd4life
i have not understand this part clearly

please help how u got this $\displaystyle {\frac{1}{2}\sin{2x}}$

$\displaystyle \frac{\cos^2{x}}{\sin{x}\cos{x}} - \frac{\sin^2{x}}{\sin{x}\cos{x}} = \frac{\cos^2{x}-\sin^2{x}}{\sin{x}\cos{x}} = \frac{\cos{2x}}{\frac{1}{2}\sin{2x}}$
• Mar 20th 2009, 01:02 AM
manutd4life
Thank you very much