Given x lies in the first quadrant i.e. 0 < x < π/2 And cos x = 8/17

We know that

LHS

= cos(30 + x) + cos(45 – x) + cos(120 – x)

We know that cos(A ±B) = cosA cosB ∓ sinA sinB

= cos 30° cos x – sin 30° sin x + cos 45° cos x + sin 45° sin x + cos 120° cos x + sin 120° sin x

= cos x(cos 30° + cos 45° + cos 120°) + sin x(-sin 30° + sin 45° + sin 120°)

= RHS

Hence, proved.