Find the exact value of tan( ( + )

cos = 2/5 3/2 < < 2 sin = -5/13 < < 3/2

  (1500 -338√21)/51

Step-by-step explanation:

The tangent is related to the cosine by ...

  tan(x) = ±√(1/cos(x)^2 -1)

For a fourth-quadrant angle, the tangent will be negative. So,

  tan(α) = -√((5/2)^2 -1) = -(√21)/2

The tangent is related to the sine by ...

  tan(x) = ±sin(x)/√(1 -sin(x)^2)

For a third-quadrant angle, the tangent will be positive, So,

  tan(β) = (5/13/)√(1 -(5/13)^2) = (5/13)/(12/13) = 5/12

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The identity for the tangent of the sum of angles tells us ...

  tan(α+β) = (tan(α) +tan(β))/(1 -tan(α)tan(β))

  = (-√21)/2 +5/12)/(1 -(-√21)/2·5/12) = ((-12√21 +10)/24)/(24+5√21)/24)

  = (10 -12√21)(24 -5√21)/(24^2 -25·21)

  tan(α+β) = (1500 -338√21)/51