Question

If y = sin(x/2) |[1/{cos(x/2) cosx}] + [1/{cosx cos(3x/2)}] + [1/{cos (3x/2) cos2x}]| then (dy/dx)_x=(π/2) =________

(a) (3/2) 

(b) (1/2) 

(c) – 1 

(d) 1

Answer 1

The correct option (d) 1

Explanation:

y = sin (x/2) [{1/(cos (x/2) ∙ cos x)} + {1/(cos x ∙ cos (3x/2))} + {1/(cos (3x/2) ∙ cos2x)}]

Consider

[{sin (x/2)}/{cos (x/2) ∙ cos x}] = [(sin {x – (x/2)})/(cos {x/2} ∙ cos x)]

= [{sinx cos(x/2) – cos x ∙ sin (x/2)}/{cos(x/2) ∙ cosx}] = tanx – tan(x/2)

∴ Similarly,

[(sin x/2)/(cos x ∙ cos {3x/2})] = tan (3x/2) – tan x and

[(sinx/2)/(cos(3x/2) ∙ cos3x)] = tan2x – tan(3x/2)

∴ We can write,

y = tanx – tan(x/2) + tan (3x/2) – tanx + tan2x – tan(3x/2)

y = tan2x – tan(x/2)

∴ (dy/dx) = sec²2x(2) – sec²(x/2) (1/2)

∴ (dy/dx)|_{at x=(π/2)} = 2sec² (π) – (1/2) sec²(π/4)

= 2 × 1 – (1/2) × 2

= 2 – 1

= 1