Hard Trigonometry expression question

Tan2X = -b/a ; [Pi/2 is less than or equal to 2x, and 2x less than or equal to Pi]

Question: Determine and expression for SinXCosX in terms of A&B.

... sin x cos x

= (1/2) ( 2 sin x cos x )

= (1/2) ( sin 2x )

= ( 1/2) ( sin 2x / cos 2x ) ( cos 2x )

= (1/2) ( tan 2x ) ( 1 / sec 2x )

= (1/2) ( tan 2x ) / √( sec² 2x )

= (1/2) ( tan 2x ) / √( 1 + tan² 2x ), ... where tan 2x = -b/a

= (1/2) ( -b/a) / √[1+(b²/a²)]

= (1/2) (-b/a) / [ √(a²+b²) / a ]

= -b / [ 2√(a²+b²) ] ............................. Ans.
_________________________

tan(2x) = -b/a

sin(2x)/cos(2x) = -b/a

2*sin(x)*cos(x) = -b/a*cos(2x)

sin(x)*cos(x) = -b/(2a) * cos(2x)

cos²(2x) = 1/(1 + b²/a²) = a²/(a² + b²)

cos(2x) = a/√(a² + b²)

sin(x)*cos(x) = -b/(2√(a² + b²))