Question

If x = sqrt(a^sin-1t), y = sqrt(a^cos-1t) show that dy/dx = -y/x

Answer 1

x = √a∧sin-1t ⇒ x = a∧sin-1t /2
y = √a∧cos-1t ⇒y = a∧cos-1t /2         ..............[∵√p = p∧(1/2)]
now, xy= (a∧sin-1t /2) × (a∧cos-1t /2)
 as bases are same(i.e. 'a') powers will be added
 ⇒xy = a∧ (sin-1t + cos-1t/2)
⇒xy  = a∧(π/4) ....................trigonometric identity[ sin-1t + cos-1t = π/2]
Differentiating both sides
⇒ xdy/dx + y = 0 ...................[d(constant)/dx = 0]
⇒xdy/dx = -y
⇒dy/dx = -y/x
    Hence Proved

Answer 2

Answer:

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