Question

The value of (tan1° tan2° tan3° ... tan89°) is ______.

विकल्प

• 0

• 1

• 2

• `1/2`

Answer 1

The value of (tan1° tan2° tan3° ... tan89°) is 1.

Explanation:

tan 1°.tan 2°.tan 3° ...... tan 89°

= tan 1°.tan 2°.tan 3°...tan 43°.tan 44°.tan 45°.tan 46°.tan 47°...tan 87°.tan 88°.tan 89°

Since, tan 45° = 1,

= tan 1°.tan 2°.tan 3°...tan 43°.tan 44°.1.tan 46°.tan 47°...tan 87°.tan 88°.tan 89°

= tan 1°.tan 2°.tan 3°…tan 43°.tan 44°.1.tan(90° – 44°).tan(90° – 43°) ...tan(90° – 3°).tan(90° – 2°).tan(90° – 1°)

Since, tan(90° – θ) = cot θ,

= tan 1°.tan 2°.tan 3°...tan 43°.tan 44°.1.cot 44°.cot 43°...cot 3°.cot 2°.cot 1°

Since, tan θ = `(1/cot θ)`

= `tan1^circ * tan2^circ * tan3°...tan43^circ * tan44^circ * 1 *  (1/tan 44^circ)`. `(1/tan 43^circ) ... (1/tan 3^circ) * (1/tan 2^circ) * (1/tan 1^circ)`

= `(tan 1^circ xx 1/tan1^circ) * (tan 2^circ xx 1/tan 2^circ) ... (tan 44^circ xx 1/tan 44^circ)`

= 1

Hence, tan 1°.tan 2°.tan 3° ...... tan 89° = 1