Answer:
(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
