Simplify tan(a+b)

Problem

tan(a+b)

Solution

1. Use the sine and cosine ratio to rewrite the tangent function in terms of sine and cosine.

tan(a+b)=sin(a+b)/cos(a+b)

2. Apply the sum identities for sine and cosine to expand the numerator and the denominator.

sin(a+b)=sin(a)*cos(b)+cos(a)*sin(b)

cos(a+b)=cos(a)*cos(b)−sin(a)*sin(b)

3. Substitute the expansions back into the fraction.

tan(a+b)=(sin(a)*cos(b)+cos(a)*sin(b))/(cos(a)*cos(b)−sin(a)*sin(b))

4. Divide every term in the numerator and denominator by cos(a)*cos(b) to express the result in terms of tangent.

tan(a+b)=((sin(a)*cos(b))/(cos(a)*cos(b))+(cos(a)*sin(b))/(cos(a)*cos(b)))/((cos(a)*cos(b))/(cos(a)*cos(b))−(sin(a)*sin(b))/(cos(a)*cos(b)))

5. Simplify the ratios using the identity tan(x)=sin(x)/cos(x)

tan(a+b)=(tan(a)+tan(b))/(1−tan(a)*tan(b))

Final Answer

tan(a+b)=(tan(a)+tan(b))/(1−tan(a)*tan(b))

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