# How can I verify the identity when it doesn't match?

• Apr 1st 2009, 05:41 AM
captaintoast87
How can I verify the identity when it doesn't match?
The questions is
Verify the identity: sec(x+y)=secxsecy/1-tanxtany

I have a list of identities but it doesn't fully match either of them. Since sec is the reciprocal of sin, can I used this: sin(x-y)=sinxcosy-cosxsiny? If so, what would take the place of cos? csc? Thanks
• Apr 1st 2009, 07:04 AM
Krizalid
Quote:

Originally Posted by captaintoast87

Since sec is the reciprocal of sin

... of cos.
• Apr 1st 2009, 07:50 AM
running-gag
Quote:

Originally Posted by Krizalid
... of cos.

...of course ! (Rofl)
• Apr 1st 2009, 11:43 AM
Soroban
Hello, captaintoast87!

Quote:

Verify the identity: .$\displaystyle \sec(x+y) \:=\:\frac{\sec x\sec y}{1-\tan x\tan y}$

The right side is: .$\displaystyle \frac{\dfrac{1}{\cos x}\cdot\dfrac{1}{\cos y}} {1 - \dfrac{\sin x}{\cos x}\cdot\dfrac{\sin y}{\cos y}}$

Multiply top and bottom by $\displaystyle \cos x\cos y$

. . $\displaystyle \frac{\cos x\cos y\left(\dfrac{1}{\cos x}\cdot\dfrac{1}{\cos y}\right)} {\cos x\cos y\left(1 - \dfrac{\sin x}{\cos x}\cdot\dfrac{\sin y}{\cos y}\right)} \;=\;\frac{1}{\underbrace{\cos x\cos y - \sin x\sin y}_{\text{This is }\cos(x+y)}}$ .$\displaystyle = \;\frac{1}{\cos(x+y)} \;=\;\sec(x+y)$