Question

If tan 15° = `2 - sqrt(3)`, what is the value of tan 15° cot 75° + tan 75° cot 15°.

विकल्प

• 8

• 14

• 12

• 10

Answer 1

14

Explanation:

tan 15° = `2 - sqrt(3)`

∴ tan (90° – 75°) = `2 - sqrt(3)`  .....`[(tan(90^circ - theta)),(= cot theta)]`

⇒ cot 75° = `2 - sqrt(3)`

⇒ `1/(tan 75^circ) = 2 - sqrt(3)`

Rationalizing the equation, we get,

⇒ tan 75° = `1/(2 - sqrt(3)) xx (2 + sqrt(3))/(2 + sqrt(3))`

⇒ tan 75° = `(2 + sqrt(3))/1 = (2 + sqrt(3))`

∴ cot 15° = `(2 + sqrt(3))`

Then value of tan15° cot75° + tan75° cot15°

= `(2 - sqrt(3))(2 - sqrt(3)) + (2 + sqrt(3))(2 + sqrt(3))`

= `4 + 3 - 4sqrt(3) + 4 + 3 + 4sqrt(3)`

= 14