Question

-cotθ tan(90 - θ)+cosecθ sec (90 - θ) + sin²37 + sin²53/tan10 tan20 tan30 tan70 tan80. Evaluate it.

Answer 1

HELLO DEAR,

Let theta = A, for simplicity in solution.

Numerator 
= -cotA tan(90 - A) +  Sec A cosec(90-A) + sin² 37 + sin² 53

= - cotA * cot A + cosec A * cosec A + sin² 37 + Cos² (90-37)

= 1 + 1

= 2

as  sin²A + cos²A = 1  for any angle A.
sec²A - tan²A = 1   for any angle A
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 30  tan 20 tan 70
 = tan 10 * 1/tan10  * tan 30 *  tan 20 * 1/tan20
 = tan 30
 = 1/√3

Answer = 2√3

I HOPE ITS HELP YOU DEAR,
THANKS