Trig Identities

tsaephenix · June 14th 2007, 11:37 AM

Okay, this is all C3 (advanced maths, the English A-Level), and I need some help.

I need to prove that sin/cos + cos/sin = 2cosec2x

... any ideas?

I got this so far:

2cosec2x

2/sin2x

2/2sinxcosx (2's cancel out)

then from there, I think I'm stuck...

Help!

Luv me x x x

**Soroban** · June 14th 2007, 11:51 AM

Hello, tsaephenix!

Prove that: . $\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} \:= \:2\csc2x$

Generally, we start with the more complicated side
. . and try to simplify it to match the other side.

The left side is: . $\tan x + \cot x \:=\:\tan x + \frac{1}{\tan x} \;=\;\frac{\tan^2\!x+1}{\tan x} \;=\;\frac{\sec^2\!x}{\tan x}$

Then we have: . $\frac{\frac{1}{\cos^2x}}{\frac{\sin x}{\cos x}} \:=\:\frac{1}{\sin x\cos x}\;=\;\frac{2}{2\sin x\cos x} \;=\;\frac{2}{\sin2x} \;=\;2\csc2x$

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