Choose the correct answer

We are given that,

We need to find the value of x^{2}.

Take,

Multiply on both sides by tangent.

Since, we know that tan(tan^{-1} x) = x.

So,

Or

Now, we need to simplify it in order to find x^{2}. So, rationalize the denominator by multiplying and dividing by .

Note the denominator is in the form: (x + y)(x – y), where

(x + y)(x – y) = x^{2} – y^{2}

So,

…(i)

Numerator:

Applying the algebraic identity in the numerator, (x – y)^{2} = x^{2} + y^{2} – 2xy.

We can write as,

Again using the identity, (x + y)(x – y) = x^{2} – y^{2}.

…(ii)

Denominator:

Solving the denominator, we get

…(iii)

Substituting values of Numerator and Denominator from (ii) and (iii) in equation (i),

By cross-multiplication,

⇒ x^{2} tan α = 1 – √(1 – x^{4})

⇒ √(1 – x^{4}) = 1 – x^{2} tan α

Squaring on both sides,

⇒ [√(1 – x^{4})]^{2} = [1 – x^{2} tan α]^{2}

⇒ 1 – x^{4} = (1)^{2} + (x^{2} tan α)^{2} – 2x^{2} tan α [∵, (x – y)^{2} = x^{2} + y^{2} – 2xy]

⇒ 1 – x^{4} = 1 + x^{4} tan^{2} α – 2x^{2} tan α

⇒ x^{4} tan^{2} α – 2x^{2} tan α + x^{4} + 1 – 1 = 0

⇒ x^{4} tan^{2} α – 2x^{2} tan α + x^{4} = 0

Rearranging,

⇒ x^{4} + x^{4} tan^{2} α – 2x^{2} tan α = 0

⇒ x^{4} (1 + tan^{2} α) – 2x^{2} tan α = 0

⇒ x^{4} (sec^{2} α) – 2x^{2} tan α = 0 [∵, sec^{2} x – tan^{2} x = 1 ⇒ 1 + tan^{2} x = sec^{2} x]

Taking x^{2} common from both terms,

⇒ x^{2} (x^{2} sec^{2} α – 2 tan α) = 0

⇒ x^{2} = 0 or (x^{2} sec^{2} α – 2 tan α) = 0

But x^{2} ≠ 0 as according to the question, we need to find some value of x^{2}.

⇒ x^{2} sec^{2} α – 2 tan α = 0

⇒ x^{2} sec^{2} α = 2 tan α

In order to find the value of x^{2}, shift sec^{2} α to Right Hand Side (RHS).

Putting ,

⇒ x^{2} = 2 sin α cos α

Using the trigonometric identity, 2 sin x cos x = sin 2x.

⇒ x^{2} = sin 2α

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