Trigonometry f(x)

Question

let f(x) = sin³x + cos³x (tan x)

 

Let sin x = 2/3

 

Show that f(2x) = - (4√5) ÷ 9

 

Thank you for your time!

Answer 1

Evaluate f(x) when x=2x.   Also, we will use the identities that are appropriate here

 

f(2x) = sin³(2x) + cos³(2x)[sin(2x)/cos(2x)]

 

f(2x) = sin³(2x) + cos²(2x)sin(2x)

 

f(2x) = sin(2x)[sin²(2x) + cos²(2x)]

 

f(2x) = sin(2x)

 

f(x) = 2sin(x)cos(x)

 

f(x) = 2sin(x)√(1 - (sin²(x))

 

 

Use the fact that sin(x)=2/3.

 

f(2x) = 2(2/3)√(1 - (4/9))

 

f(2x) = (4/3)√(5/9)

 

f(2x) = (4/3)√(5) / 3

 

f(2x) = 4√(5) / 9

 

 

Now I have a question for you.  Where did you get the negative sign in your answer from?