Question

Evaluate sec theta cosec 90 degree minus theta minus 10 theta cot 90 degree minus theta + sin square 35 degree + sin square 55 degree / tan10 degree tan20 degree tan45 degree tan70 degree tan80 degree

Answer 1

Let theta = A, for simplicity in solution.

Numerator
= Sec A cosec(90-A) - Tan A Cot(90 - A) + sin² 55 + sin² 35
= Sec A * sec A - tan A * tan A + sin² 55 + Cos² (90-35)
= 1 + 1
= 2

as sin²A + cos²A = 1 for any angle A.
sec²A - tan²A = 1 for any angle
cosec(90-A) = sec A
cot (90-A) = tan A
=====
we know that tan A = cot (90-A) = 1 / tan (90-A) . Let us apply this.

Denominator
= tan 10 tan 80 tan 60 tan 20 tan 70
= tan 10 * 1/tan10 * tan 60 * tan 20 * 1/tan20
= tan 60
= √3

Answer = 2/√3