Question

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Class 12!!

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Answer 1

EXPLANATION.

⇒ tan⁻¹x + tan⁻¹y = 4π/5.

As we know that,

Formula of :

⇒ tan⁻¹x + cot⁻¹x = π/2.

Using this formula we can write equation as,

⇒ tan⁻¹x = π/2 - cot⁻¹x.   -------(1).

⇒ tan⁻¹y + cot⁻¹y = π/2.

⇒ tan⁻¹y = π/2 - cot⁻¹y.  --------(2).

From equation (1) & (2), we get.

⇒ π/2 - cot⁻¹x + π/2 - cot⁻¹y = 4π/5.

⇒ π/2 + π/2 - cot⁻¹x - cot⁻¹y = 4π/5.

⇒ 2π/2 - 4π/5 = cot⁻¹x + cot⁻¹y.

⇒ π - 4π/5 = cot⁻¹x + cot⁻¹y.

⇒ 5π - 4π/5 = cot⁻¹x + cot⁻¹y.

⇒ π/5 = cot⁻¹x + cot⁻¹y.

⇒ cot⁻¹x + cot⁻¹y = π/5.

Option [A] is correct answer.

                                                                                                                     

MORE INFORMATION.

Properties of trigonometric inverse function.

(1) = sin⁻¹(-x) = -sin⁻¹x.

(2) = cos⁻¹(-x) = π - cos⁻¹x.

(3) = tan⁻¹(-x) = -tan⁻¹(x).

(4) = cot⁻¹(-x) = π - cot⁻¹x.

(5) = sec⁻¹(-x) = π - sec⁻¹x.

(6) = cosec⁻¹(-x) = -cosec⁻¹x.

(1) = sin⁻¹x + cos⁻¹x = π/2.

(2) = tan⁻¹x + cot⁻¹x = π/2.

(3) = sec⁻¹x + cosec⁻¹x = π/2.

Answer 2

$\sf\underline\pink{Given}$

$⇒ tan⁻¹x + tan⁻¹y = \sf\tt\frac{4π}{5}$

$\sf\underline\red{Solution}$

As we know that,

Formula of :

$⇒ tan⁻¹x + cot⁻¹x = \sf\frac{π}{2}$

Using this formula we can write equation as,

$⇒ tan⁻¹x = \sf\frac{π}{2} - cot⁻¹x----(1).$

$⇒ tan⁻¹y + cot⁻¹y = \sf\frac{π}{2}.$

$⇒ tan⁻¹y = \sf\frac{π}{2}- cot⁻¹y----(2).$

From equation (1) & (2), we get.

$⇒ \sf\frac{π}{2} - cot⁻¹x + \sf\frac{π}{2}- cot⁻¹y = \sf\frac{4π}{5}$

$⇒ \sf\frac{π}{2} + \sf\frac{π}{2} - cot⁻¹x - cot⁻¹y = \sf\frac{4π}{5}$

$⇒\sf\frac{2π}{2} - \sf\frac{4π}{5} = cot⁻¹x + cot⁻¹y.$

$⇒ π - \sf\frac{4π}{5} = cot⁻¹x + cot⁻¹y.$

$⇒ 5π - \sf\frac{4π}{5} = cot⁻¹x + cot⁻¹y.$

$⇒ \sf\frac{π}{5} = cot⁻¹x + cot⁻¹y.$

$⇒ cot⁻¹x + cot⁻¹y = \sf\frac{π}{5}.$

$\sf\underline\purple{Option [A] is correct answer.}$

                                                                                                                     

MORE INFORMATION-

Properties of trigonometric inverse function.

1) = sin⁻¹(-x) = -sin⁻¹x.

2) = cos⁻¹(-x) = π - cos⁻¹x.

3) = tan⁻¹(-x) = -tan⁻¹(x).

4) = cot⁻¹(-x) = π - cot⁻¹x.

5) = sec⁻¹(-x) = π - sec⁻¹x.

6) = cosec⁻¹(-x) = -cosec⁻¹x.

1) = sin⁻¹x + cos⁻¹x = π/2.

2) = tan⁻¹x + cot⁻¹x = π/2.

3) = sec⁻¹x + cosec⁻¹x = π/2.