Question

Evaluate:sec theta.cosec (90-theta)- tan theta.cot (90-theta)+ sin^2 55°+ sin^2 °35/tan10°.tan20°.tan60°.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