How Do You Prove Trigonometric Identities Involving Double Angles and Tangents?

[ku1005]

Hi, in this question i am nt sure the best way to tackle it!
it follows

proove the following

2sinxcosx=sqrt(3)-ssqrt(3)sin^2x for 0<=x<=360

i tried using the doble angle formulae on the right, putting all on one side therefore =0 (anticipating a quadratic equation)
having

sin2x-sqrt(3)+2sqrt(3)sin^2(x)

i can see that a quadratic equation is smhow possible, but don't know how to get it there...any help or tips would be greatly apprecitaed!

thanks!

 

[ku1005]

actually the double angle formulae on the left...

 

[Pyrrhus]

is the identity?

$$2 \sin x \cos x = \sqrt{3} - \sqrt{3} \sin^{2} x$$

because the above equation is not an identity.

 

[ku1005]

sorry...not an identity...was readin the wrong stuff...it just wants me to solve for x

 

[Integral]

I don't think you need to go that route.

factor $\sqrt {3}$ from the RHS. Do you see anything that looks familiar?

 

[ku1005]

u mean how the (1-2sin^2x) becomes (1-2(1-cos^2x)?
hang on i will see how that works

 

[ku1005]

which then looks like double angle formulae for cos

 

[ku1005]

gerat thanks very muc...get it down to tan2x=sqrt(3) thanks for ur help

 

[Integral]

Did you get all of the solutions?

My last question was in reference to the ORIGINAL equation. You do not need to use a double angle relationship to solve this.

 

[ku1005]

ohh kk...dunno um i got all the soltutions... so thanks, also this is a real common identity whih i am trying to proove

sin2x=2tanx/(tan^2x+1)

i am trying yo simplify the RHS,but evertyhing i do makes it more complicated...i must be missing somthing simple...hat should i start with??

 

[Pyrrhus]

ohh kk...dunno um i got all the soltutions... so thanks, also this is a real common identity whih i am trying to proove

sin2x=2tanx/(tan^2x+1)

i am trying yo simplify the RHS,but evertyhing i do makes it more complicated...i must be missing somthing simple...hat should i start with??

one huge hint:

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