Question

show that 1/sin10-√3/cos10=4

Answer 1

1/sin10°-√3/cos10°
=(cos10°-√3sin10°)/sin10°cos10°
=[2{(1/2)cos10°-(√3/2)sin10°}]/[(1/2)(2sin10°cos10°)]
=[(2×2)(sin30°cos10°-cos30°sin10°)]/sin20°
[∵, sin30°=1/2, cos30°=√3/2, 2sinФcosФ=sin2Ф]
=4sin(30°-10°)/sin20°
[∵, sinαcosβ-cosαsinβ=sin(α-β)]
=4sin20°/sin20°
=4 (Proved)