Question

Evaluate tan15 + sin25 cot75 cos65

Answer 1

we know  cot 75 = tan (90 - 75) = tan 15
                 cos 65 = sin (90- 65) = sin 25

LHS = Tan 15 +  sin 25 cot 75  cos 65
  =  Tan 15 (1 + Sin² 25)
  =  0.268 * (1 + 0.422²) 
  =  0.316

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Find cos 15, tan 15   by  solving polynomial equations or by using calculator.

Cos 30 = √3 /2
             = 2 cos²15° - 1

=> cos² 15° = [2+√3] / 4          and  Cos 15 = √2 [√3 +1] / 4
=> Sin² 15° = [2-√3] /4           and   Sin 15 = √2[√3 - 1] / 4
=> tan² 15° = [2-√3] / [2+√3]
                   =  [2 -√3]²    by rationalization.
=>  Tan 15 = [2-√3]  = 0.268

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Let x = cosA   and   A = 65°   
4 cos³A - 3 cosA  = cos 3A
4 cos³ 65 - 3 cos 65 = cos 195 = - cos 15 = -√2 [√3+1]/4
x³ - 3/4 x + √2 [√3+1] /16  = 0
Let   x = y + 1 / (4y) 
Using cubic equation solving methods we get:
     y³ = (√6+√2)/32  +- [√(12+2√3)  /2]

  We find  y³ and then y.
  Then we find x and then cos 65 = sin 25 = x
   x = 0.422