Question

tan(9)-tan(27)-tan(63)+tan(81)

Answer 1

Answer:

4

Step-by-step explanation:

tan(9)-tan(27)-tan(63)+tan(81)

tan(81) + tan(9) - (tan(63) + tan(27))

by identity tan 81 = cot 9, tan 63 = cot 27

cot(9) + tan(9) - (cot(27) + tan(27))

(cos(9) / sin(9) + sin(9) / cos(9)) - (cos(27) / sin(27) + sin(27) / cos(27))

((cos(9)cos(9) + sin(9)sin(9)) / sin(9)cos(9)) - ((cos(27)cos(27) + sin(27)sin(27)) / sin(27)cos(27))

((cos²(9) + sin²(9) /  sin(9)cos(9)) - ((cos²(27) + sin²(27) /  sin(27)cos(27))

by identity cos² x + sen² x = 1

1 / sin(9)cos(9) - 1 / sin(27)cos(27)

by identity sin(2x) = 2sin(x)cos(x)

sin(2x) / 2 = sinxcos(x)

1 / (sin(2*9) / 2) - 1 / (sin(2*27) / 2)

2 / sin(18) - 2 / sin(54)

2 ((sin(54) - sin(18)) / sin(54)sin(18))

by identity sin(3x) - sin(x) = 2cos(2x)sen(x)

2(2cos(36)sin(18)) / cos(36)sin(18)

4