Find the value of $\dfrac{sin (-660^o) tan (1050^o) sec (-420^o)}{cos (225^o ) cosec (315^o) cos(510^o)}$
- $\dfrac{\sqrt 3}{4}$
- $\dfrac{\sqrt 3}{2}$
- $\dfrac{2}{\sqrt 3}$
- $\dfrac{4}{\sqrt 3}$
Simplify each term: sin(-660) = sin(60) = sqrt(3)/2. tan(1050) = tan(330) = -1/sqrt(3). sec(-420) = sec(60) = 2. cos(225) = -1/sqrt(2). cosec(315) = -sqrt(2). cos(510) = cos(150) = -sqrt(3)/2. Numerator: (sqrt(3)/2) * (-1/sqrt(3)) * 2 = -1. Denominator: (-1/sqrt(2)) * (-sqrt(2)) * (-sqrt(3)/2) = -sqrt(3)/2. Result: -1 / (-sqrt(3)/2) = 2/sqrt(3).
Using the odd and even function properties along with reference angles, the numerator simplifies from sin negative 660 degrees times tan 1050 degrees times sec negative 420 degrees to sin 60 degrees times tan 60 degrees times sec 60 degrees. This evaluates to the square root of 3 over 2 times the square root of 3 times 2, giving 3. Applying the same angle reduction rules to the denominator yields cos 225 degrees times cosec 315 degrees times cos 510 degrees, which simplifies to negative the square root of 2 over 2 times negative the square root of 2 times the square root of 3 over 2. Multiplying the denominator terms gives 3 times the square root of 3 over 2, so dividing the numerator by the denominator results in 2 over the square root of 3.