Question

Evaluate tan50.tan25.tan65.tan45.tan85=?

Answer 1

Tan 25  =  Cot (90 - 25) = cot  65
So  Tan 25 * Tan 65 =  1

We know   Tan 45 = 1

Tan 85 = Cot (90 - 85) = Cot 5

so LHS seems to be  equal to tan 50 tan 85 = cot 40 cot 5 = tan 50 / tan 5

i have not got an expression in terms of rational or irrational numbers...
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tan 50 * tan 85
 = cot 40 cot 5
 = [cos 40 cos 5] / [sin 40 sin 5]
 = [cos 45 + cos 35] /[ cos 35 - cos 45]
 = [1 + √2 cos 35] / [ √2 cos 35 - 1 ]
 = [ 1 + 2 cos² 35 + 2√2 cos 35 ] / [2 cos² 35 -1 ]
 =  [ 2 + cos 70 + 2 √2 cos 35 ] / cos 70
 = 1 + 2 sec 70 +  2√2 Cos 35  sec 70
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Tan 50 * Tan 85 =  Tan (45 + 5) * Cot 5
         =  [ tan 45 + tan 5] / [ 1 - tan 45 * tan 5 ]   *  cot 5
         =  [cot 5 + 1] / [ 1 - tan 5 ]
         = (1 + tan 5) / [tan 5 (1 - tan 5)]
         =   cot 5 * [ 2  / (1 - tan 5)  - 1 ]  
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tan 50 * tan 85 =
= (cos 5 + sin 5) cos 5  /  [ sin 5 ( cos 5 - sin 5) ]
= [ cos^2 5 + sin 10 /2 ]  /  [ sin 10 /2 - sin^2 5 ]
= [ 1 + cos 10 + sin 10 ] / [ sin 10 + cos 10 - 1]
= [ 1 + cos 10 + sin 10 ][ 1 + cos 10 + sin 10 ] / [ 1 + cos 10 + sin 10 ]* [ sin 10 + cos 10 - 1]
= [1+2cos10 +2sin10+1+sin20] / sin20
= [ (cos10+sin10)²+2 (cos10+sin10) ]/ sin 20
= (cos10+sin10)(cos10+sin10+2)/sin20
= (1+tan10)(cot10+1+2cosec10)/2
= [1 + 2 (sin80+sin10) + sin 20 ] / sin 20
= [1 + 4 Sin 45 Cos35 + sin 20 ] / sin 20