Tangent Sum Identity
tan(A+B)=tanA+tanB/1-tanAtanB
The tangent sum identity is a trigonometric identity that relates the tangent function of the sum of two angles to the tangent functions of the individual angles. It is given by:
tan(a + b) = (tan a + tan b) / (1 – tan a tan b)
where a and b are any two angles such that tan a and tan b are defined and not equal to ±1.
The proof of the tangent sum identity involves using the fundamental trigonometric identity for tangent, which states that:
tan x = sin x / cos x
Using this identity and some algebraic manipulations, we can write:
tan(a + b) = sin(a + b) / cos(a + b)
= (sin a cos b + cos a sin b) / (cos a cos b – sin a sin b)
= (sin a / cos a + sin b / cos b) / (1 – sin a / cos a * sin b / cos b)
= (tan a + tan b) / (1 – tan a tan b)
Thus, the tangent sum identity follows from the fundamental trigonometric identity for tangent and some algebraic manipulations. This identity finds many applications in solving various trigonometric equations and problems.
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