Derivation of Tangent Addition
The formula for tangent addition is derived by using the formulas for expansion of sum angle for sine and cosine ratios.
Now, we know that,
tan (A + B) = sin (A + B)/cos (A + B) …… (1)
Substitute sin (A + B) = sin A cos B + cos A sin B and cos (A + B) = cos A cos B – sin A sin B in the equation (1).
tan (A + B) = (sin A cos B + cos A sin B)/(cos A cos B – sin A sin B)
Dividing the numerator and denominator by cos A cos B, we get
tan (A + B) = [(sin A cos B + cos A sin B)/(cos A cos B)]/[(cos A cos B – sin A sin B)/(cos A cos B)]
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)]
tan (A + B) = (tan A + tan B)/(1 – tan A tan B)
This derives the formula for tangent addition of any two angles, A and B.
Tangent Addition Formula
Trigonometric identities are equalities using trigonometric functions that hold true for any value of the variables involved, hence defining both sides of the equality. These are equations that relate to various trigonometric functions and are true for every variable value in the domain. The formulae sin(A+B), cos(A-B), and tan(A+B) are some of the sum and difference identities.
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