Choose the correct answer

If , then the value of x is

We are given that,

…(i)

We need to find the value of x.

Using the property of inverse trigonometry,

Replace A by and B by .

Putting this value in equation (i),

Taking tangent on both sides,

Using the property of inverse trigonometry,

tan(tan^{-1} A) = A

Cross-multiplying, we get

Simplifying the equation in order to find the value of x,

Let us cancel the denominator from both sides of the equation.

⇒ x(x + 1) + (x – 1)(x – 1) = -7[x(x – 1) – (x + 1)(x – 1)]

⇒ x^{2} + x + (x – 1)^{2} = -7[x^{2} – x – (x + 1)(x – 1)]

Using the algebraic identity,

(a – b) = a^{2} + b^{2} – 2ab

And, (a + b)(a – b) = a^{2} – b^{2}

⇒ x^{2} + x + x^{2} + 1 – 2x = -7[x^{2} – x – (x^{2} – 1)]

⇒ 2x^{2} – x + 1 = -7[x^{2} – x – x^{2} + 1]

⇒ 2x^{2} – x + 1 = -7[1 – x]

⇒ 2x^{2} – x + 1 = -7 + 7x

⇒ 2x^{2} – x – 7x + 1 + 7 = 0

⇒ 2x^{2} – 8x + 8 = 0

⇒ 2(x^{2} – 4x + 4) = 0

⇒ x^{2} – 4x + 4 = 0

We need to solve the quadratic equation to find the value of x.

⇒ x^{2} – 2x – 2x + 4 = 0

⇒ x(x – 2) – 2(x – 2) = 0

⇒ (x – 2)(x – 2) = 0

⇒ x = 2 or x = 2

Hence, x = 2.

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