Question

If 2y = [cot⁻¹(√3 cos x + sin x)/(cos x - √3 sin x)]², x ∈ (0, π/2) then dy/dx
is equal to
(A) (π/6) - x
(B) (π/3) - x
(C) x - (π/6)
(D) 2x
- (π/3)

Answer 1

$\huge{\blue{\boxed{\boxed{\boxed{\pink{\underline{\underline{\mathfrak{\red{♡ĂnSwer♡}}}}}}}}}}$

$2y = ( \cot {}^{ - 1} ( \frac{ \sqrt{3} \cos(x) + \sin(x) }{ \cos(x) - \sqrt{3} \sin(x) } ) ) {}^{2}$

$2y = ( \cot {}^{ - 1} ( \frac{ \sqrt{3} + \tan(x) }{1 - \sqrt{3} \tan(x) } )) {}^{2}$

$2y = (\cot {}^{ - 1} ( \frac{ \tan( \frac{\pi}{3} ) + \tan(x) }{1 - \tan( \frac{\pi}{3} ) \tan(x) } ) ) {}^{2}$

$2y = ( \cot {}^{ - 1} ( \tan( \frac{\pi}{3} + x ) ) {}^{2}$

we know that

tan⁻¹x + cot⁻¹x = π/2

___________________

then ,

$2y = ( \frac{\pi}{2 } - \tan {}^{ - 1} ( \tan( \frac{\pi}{3} + x ) ) ) {}^{2}$

$2y = ( \frac{\pi}{2} - \frac{\pi}{3} - x) {}^{2}$

$2y = ( \frac{\pi}{6} - x) {}^{2}$

$2y = \frac{\pi {}^{2} }{36} + x {}^{2} - \frac{\pi \: x}{3}$

now differinate with respect to x

$2 \times \frac{dy}{dx} = 2x - \frac{\pi}{ 3}$

$\frac{dy}{dx} = x - \frac{\pi}{6}$

correct option c)

Answer 2

2y=(cot

−1

(

cos(x)−

3

sin(x)

3

cos(x)+sin(x)

))

2

2y = ( \cot {}^{ - 1} ( \frac{ \sqrt{3} + \tan(x) }{1 - \sqrt{3} \tan(x) } )) {}^{2}2y=(cot

−1

(

1−

3

tan(x)

3

+tan(x)

))

2

2y = (\cot {}^{ - 1} ( \frac{ \tan( \frac{\pi}{3} ) + \tan(x) }{1 - \tan( \frac{\pi}{3} ) \tan(x) } ) ) {}^{2}2y=(cot

−1

(

1−tan(

3

π

)tan(x)

tan(

3

π

)+tan(x)

))

2

2y = ( \cot {}^{ - 1} ( \tan( \frac{\pi}{3} + x ) ) {}^{2}2y=(cot

−1

(tan(

3

π

+x))

2

we know that

tan⁻¹x + cot⁻¹x = π/2

___________________

then ,

2y = ( \frac{\pi}{2 } - \tan {}^{ - 1} ( \tan( \frac{\pi}{3} + x ) ) ) {}^{2}2y=(

2

π

−tan

−1

(tan(

3

π

+x)))

2

2y = ( \frac{\pi}{2} - \frac{\pi}{3} - x) {}^{2}2y=(

2

π

−

3

π

−x)

2

2y = ( \frac{\pi}{6} - x) {}^{2}2y=(

6

π

−x)

2

2y = \frac{\pi {}^{2} }{36} + x {}^{2} - \frac{\pi \: x}{3}2y=

36

π

2

+x

2

−

3

πx

now differinate with respect to x

2 \times \frac{dy}{dx} = 2x - \frac{\pi}{ 3}2×

dx

dy

=2x−

3

π

\frac{dy}{dx} = x - \frac{\pi}{6}

dx

dy

=x−

6

π

correct option c)