Express tan(2x) in terms of sin(x) alone.

[noob^n]

Homework Statement

Express tan(2x) in terms of sin(x) alone.

assuming: pi < x < 3pi/2

Homework Equations

Trig identities

The Attempt at a Solution

sin2x/cos2x

switched for double angle equations;

(2sinx*cosx)/((cosx)^2 - (sinx)^2)

then wherever i go with it, it leads nowhere.

 

[rocomath]

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

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

 

[Miss_AnnA]

Well tan2x=Sin2x/Cos2x then
Sin2x= Tan2x*Cos2x but note that Cos^2(2x)=1/Sec^2(2x) using sec^2(2x)=1+ tan^2(2x) we then get
Sin2x=Tan2x/Sqrt(1+tan^2(2x)) this is all ok but sin2x=2sinxcosx so you need to do the same for cos2x and find cosx in terms of tan2x thus replace it into the expression above.
I hope this helps.

 

[Miss_AnnA]

Yes I just realized that you can get nicer expression if you see that

cos2x=1/Sqrt(1+tan^2(2x)) then cos2x=1-2sin^2(x)
hence
1-2sin^2(x)=1/(Sqrt(1+tan^2(2x)))