Question

-tan theta•cot(90-theta)+sec theta•cosec(90-theta)+sin square35+sin square55/tan10•tan20•tan30•tan70•tan80

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