Simplify tan(4x)

Problem

tan(4*x)

Solution

1. Identify the expression as a double angle of 2*x to apply the tangent double angle identity.

2. Apply the formula for tan(2*θ) where θ=2*x

tan(4*x)=(2*tan(2*x))/(1−tan2(2*x))

3. Substitute the double angle identity for tan(2*x) into the expression, where tan(2*x)=(2*tan(x))/(1−tan2(x))

tan(4*x)=(2*((2*tan(x))/(1−tan2(x))))/(1−((2*tan(x))/(1−tan2(x)))2)

4. Simplify the numerator and the denominator by finding a common denominator for the bottom fraction.

tan(4*x)=(4*tan(x))/(1−tan2(x))/((1−tan2(x))2−4*tan2(x))/((1−tan2(x))2)

5. Multiply by the reciprocal to simplify the complex fraction.

tan(4*x)=(4*tan(x))/(1−tan2(x))⋅((1−tan2(x))2)/(1−2*tan2(x)+tan4(x)−4*tan2(x))

6. Cancel the common factor of (1−tan2(x)) and combine like terms in the denominator.

tan(4*x)=(4*tan(x)*(1−tan2(x)))/(1−6*tan2(x)+tan4(x))

Final Answer

tan(4*x)=(4*tan(x)−4*tan3(x))/(1−6*tan2(x)+tan4(x))

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