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Trigonometry

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start with the identity tan2θ + 1 = sec2θ
 
substituting, 2(sec2θ - 1) = 5 secθ + 10  giving an equation in secθ.
 
substituting x = secθ
 
2(x2 - 1) = 5x + 10,   2x2 -5x -12 = 0
 
Using the quadratic formula,  x = (5 ± √(25 + 96))/4,   or  x = 4, or x = -3/2
 
Since x = secθ,  and since the angle is obtuse (between 90 and 180 degrees),  secθ = -3/2 , so cosθ = -2/3 and sinθ = √(5/9) = √5/3,   and tanθ = -√5/2