Question

prove that tan9°-tan27°-cot27°+cot9°=4​

Answer 1

Answer:

 LHS = tan9° - tan27° - cot27° + cot9°

 = (tan9° + cot9°) - (tan27° + cot27°)

 = (sin9°/cos9° + cos9°/sin9°) - (sin27°/cos27° + cos27°/sin27°)

 = (sin²9° + cos²9°)/sin9°cos9° - (sin²27° + cos²27°)/sin27° cos27°

= 1/sin9°cos9° - 1/sin27°cos27°

 = 2/(2sin9°cos9°) - 2/(2sin27°cos27°)

 = 2/sin18° - 2/sin54° [ as we know , sin2x = 2sinxcosx]

 = 2 [ (sin54° - sin18°)/sin18°.sin54°]

use formula, sinX - sinY = 2cos(X + Y)/2sin(X - Y)/2

 = 2[(2cos36°sin18°)/sin18°sin54°]

 = 4[ cos36°/cos(90° - 36°) ]

 = 4[ cos36°/cos36° ]

 = 4

= RHS

Answer 2

Here is your answer

Hope it helps you