Question

Prove that: (sin 300)(tan 330)(sec 420) / (cot 135)(cos 210)(cosec 315)= rootof(2/3). ROOT FOR BOTH 2 AS WELL AS 3)

Answer 1

sin300°tan330°sec420°/cot135°cos210°cosec315°
=[{sin(90°×3)+30°}{tan(90°×3)+60°}{sec(90°×4)+60°}]/
[{cot(90°×1)+45°}{cos(90°×2)+30°}{cosec(90°×3)+45°}]
={(-cos30°)(-cot60°)(sec60°)}/{(-tan45°)(-cos30°)(-sec45°)}
={-√3/2×(-1/√3)×2}/{(-1)×(-√3/2)(-√2)}
=1/(-√3/√2)
=-√2/√3
The answer is: -√(2/3) √

Answer 2

sin 300 = sin (360- 60) = - sin 60 = - √3 /2
tan 330 = tan (360 - 30) = - tan 30  = - 1/√3
sec 420 = sec(360+60) = sec 60 = 2

cot 135 = cot (90+45) = - tan 45 = -1
cos 210 = cos (180+30) = - cos 30 = - √3/2
cosec 315 = cosec(360 - 45) = - cosec 45 = - √2

So   LHS=  -√3/2 * -1/√3 * 2 / [ -1 * -√3/2 * - √2 ]
              =  √(2/3)