Formula of tangent of sumFormula of tangent of sum

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll write the tangent formula:

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

We'll prove this formula substituting the sine and cosine functions by their formulas for the sine and cosine of the sum of angles a and b:

sin(a+b) = sina*cosb + sinb*cosa

cos(a+b) = cosa*cosb - sina*sinb

We'll substitute sin(a+b) and cos(a+b) by their formulas:

tan(a+b) = (sina*cosb + sinb*cosa)/(cosa*cosb - sina*sinb)

We'll factorize by cosa*cosb:

tan(a+b) =cosa*cosb*[(sina*cosb/cosa*cosb) + (sinb*cosa/cosa*cosb)]/cosa*cosb*[1 - (sina*sinb/cosa*cosb)]

We'll simplify and we'll get:

tan(a+b) = (sina/cos a + sinb/cos b)/(1 - tan a*tan b)

The formula of tangent of the sum of 2 angles is:

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