Question

Evaluate:-tan theta.cot(90-theta)+sec theta.cosec(90-theta)+sin2 35+sin2 55/tan10.tan20.tan30.tan 70.tan 80

Answer 1

$\Huge{\bold{\mathrm{\underline{\red{ANSWER}}}}}$

★ 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