Simplify tan(3x)

Problem

tan(3*x)

Solution

1. Apply the tangent sum formula by rewriting the argument 3*x as 2*x+x

tan(3*x)=tan(2*x+x)

2. Use the identity for the tangent of a sum, tan(A+B)=(tan(A)+tan(B))/(1−tan(A)*tan(B))

tan(2*x+x)=(tan(2*x)+tan(x))/(1−tan(2*x)*tan(x))

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

tan(3*x)=((2*tan(x))/(1−tan2(x))+tan(x))/(1−((2*tan(x))/(1−tan2(x)))*tan(x))

4. Find a common denominator for the numerator and the denominator of the complex fraction.

tan(3*x)=(2*tan(x)+tan(x)*(1−tan2(x)))/(1−tan2(x))/(1−tan2(x)−2*tan2(x))/(1−tan2(x))

5. Simplify the expression by canceling the common denominator 1−tan2(x) and combining like terms.

tan(3*x)=(2*tan(x)+tan(x)−tan3(x))/(1−3*tan2(x))

6. Combine the terms in the numerator to reach the final simplified form.

tan(3*x)=(3*tan(x)−tan3(x))/(1−3*tan2(x))

Final Answer

tan(3*x)=(3*tan(x)−tan3(x))/(1−3*tan2(x))

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