Final answer:
To prove the given equation involving tangent, we use trigonometric identities and simplification techniques to show that both sides of the equation are equal.
Step-by-step explanation:
To prove the given equation: tan(π/4 x) - tan(π/4-x)/tan(π/4 x) tan(π/4-x) = 2sin(x)cos(x).
We can simplify this equation using the trigonometric identities for tan:
- tan(a) = sin(a)/cos(a)
- tan(π/4) = 1
By substituting these identities, we get:
1 - 1/(tan(π/4-x)tan(π/4 x)) = 2sin(x)cos(x)
From here, we can simplify further and use the following trigonometric identities:
- sin(2a) = 2sin(a)cos(a)
- tan(a)tan(b) = sin(a+b)/cos(a+b)
Using these identities, we can simplify the equation to:
-2sin(x)cos(x) = -2sin(x)cos(x)
Therefore, the given equation is proved to be true.