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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...

=====================

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

======================

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 ]

===========================

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

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...

=====================

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

======================

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 ]

===========================

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