1. ## Idenities

Hey does anyone know these trig idenities??

((Cosx +1)/(secx-tanx)) - ((1-cosx)/secx+tanx)) = 2(1+tanx)

sec(-x) / sin(180-x) = tan(270-x) + tan(180+x)

(tanx-sinx) / (sin(cubed)x) = (sec) / (1+cosx)

2. Hello Giggly2
Originally Posted by Giggly2
Hey does anyone know these trig idenities??
If you mean do I know how to prove them, yes.
((Cosx +1)/(secx-tanx)) - ((1-cosx)/secx+tanx)) = 2(1+tanx)
For this one, you need to know that if you divide both sides of $\cos^2x +\sin^2x = 1$ by $\cos^2x$, you get:

$1+\tan^2x = \sec^2x$

$\Rightarrow \sec^2x-\tan^2x = 1$

$\Rightarrow (\sec x+\tan x)(\sec x - \tan x) = 1$

Then just put both the fractions on the LHS over a common denominator, multiply out the brackets (be careful with the minus signs!) and you'll get $2(1 + \tan x)$. I'll start you off:

$\frac{\cos x +1}{\sec x -\tan x}-\frac{\cos x -1}{\sec x +\tan x} = \frac{(\cos x +1)(\sec x +\tan x)-(1-\cos x)(\sec x - \tan x)}{(\sec x+\tan x)(\sec x - \tan x)}$

$= \frac{\dots \quad \dots \quad \dots \quad \dots}{1}$

Can you complete it now?

sec(-x) / sin(180-x) = tan(270-x) + tan(180+x)
You need to know the following identities:

• $\sec(-x) = \sec x$
• $\sin (180 - x) = \sin x$
• $\tan(180 + x) = \tan x$
• So $\tan(270 - x) = \tan(180 + 90 - x)=\tan(90-x) = \cot x$

Then $\frac{\sec(-x)}{\sin(180-x)}= \frac{\sec{x}}{\sin x} = \frac{1}{\cos x\sin x}$

Now

• start from the RHS using the last two identities above
• express $\cot x$ and $\tan x$ in terms of sine and cosine
• use $\cos^2x +\sin^2x = 1$

and you're there.

(tanx-sinx) / (sin(cubed)x) = (sec) / (1+cosx)
Again, I'll start you off:

$\frac{\tan x - \sin x}{\sin^3x}= \frac{\frac{\sin x}{\cos x}-\sin x}{\sin^3x}$

$= \frac{\sin x - \sin x \cos x}{\cos x\sin^3x}$

Now you need to:

• factorise the numerator
• divide top and bottom by the common factor $\sin x$
• use $\sin^2x = 1-\cos^2x =(1+\cos x)(1-\cos x)$
• divided top and bottom by a common factor again

You should end up with $\frac{1}{\cos x(1+\cos x)}= \frac{\sec x}{1+\cos x}$

Can you finish them off now?