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Prove tan(pi/4 x) - tan(pi/4-x)/tan(pi/4 x) tan(pi/4-x)=2sinxcosx

User Xelnor
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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.

User Skycorsarius
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