Verify the Identity (2tan(x))/(1+tan(x)^2)=sin(2x)

Problem

(2*tan(x))/(1+tan2(x))=sin(2*x)

Solution

1. Identify the Pythagorean identity for the denominator, which states that 1+tan2(x)=sec2(x)

2. Substitute the identity into the left side of the equation.

(2*tan(x))/sec2(x)

3. Rewrite the trigonometric functions in terms of sine and cosine using the definitions tan(x)=sin(x)/cos(x) and sec(x)=1/cos(x)

(2⋅sin(x)/cos(x))/1/cos2(x)

4. Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator.

2⋅sin(x)/cos(x)⋅cos2(x)

5. Cancel the common factor of cos(x) from the numerator and denominator.

2*sin(x)*cos(x)

6. Apply the double-angle formula for sine, which states that sin(2*x)=2*sin(x)*cos(x)

sin(2*x)

Final Answer

(2*tan(x))/(1+tan2(x))=sin(2*x)

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