Solve tan⁻¹(x) + tan⁻¹(y) = π/4 for given conditions

Question

If $\tan^{-1}(x) + \tan^{-1}(y) = \frac{\pi}{4}$ , where $xy < 1$ , express $y$ in terms of $x$ .

(NCERT Class 12, Chapter 2 — Inverse Trigonometric Functions)

Solution — Step by Step

When $xy < 1$ , the addition formula is:

\tan^{-1}(x) + \tan^{-1}(y) = \tan^{-1}\left(\frac{x + y}{1 - xy}\right)

This is the standard identity — the condition $xy < 1$ ensures the sum lies in $(-\pi/2, \pi/2)$ .

\tan^{-1}\left(\frac{x + y}{1 - xy}\right) = \frac{\pi}{4}

Taking $\tan$ on both sides:

\frac{x + y}{1 - xy} = \tan\frac{\pi}{4} = 1

x + y = 1 - xy

y + xy = 1 - x

y(1 + x) = 1 - x

\boxed{y = \frac{1 - x}{1 + x}}

This is valid for $x \neq -1$ and $xy < 1$ .

Why This Works

The addition formula for $\tan^{-1}$ converts a sum of two inverse tangents into a single inverse tangent with a combined argument. Setting this equal to $\pi/4$ is convenient because $\tan(\pi/4) = 1$ , which simplifies the algebra.

The result $y = (1-x)/(1+x)$ has a nice geometric interpretation: if you think of $x$ and $y$ as slopes of two lines, their combined angle (sum of the angles each makes with the x-axis) equals $45°$ .

Notice that when $x = 0$ , $y = 1$ (since $\tan^{-1}(0) + \tan^{-1}(1) = 0 + \pi/4 = \pi/4$ ). This serves as a quick verification.

Alternative Method — Direct substitution

Let $\tan^{-1}(x) = \alpha$ and $\tan^{-1}(y) = \beta$ , so $\alpha + \beta = \pi/4$ .

Then $\beta = \pi/4 - \alpha$ , and:

y = \tan\beta = \tan\left(\frac{\pi}{4} - \alpha\right) = \frac{1 - \tan\alpha}{1 + \tan\alpha} = \frac{1 - x}{1 + x}

using the $\tan(A - B)$ formula.

For JEE, the addition formula for $\tan^{-1}$ has three cases depending on $xy$ : (1) If $xy < 1$ : $\tan^{-1}x + \tan^{-1}y = \tan^{-1}\frac{x+y}{1-xy}$ . (2) If $xy > 1$ and $x > 0$ : add $\pi$ . (3) If $xy > 1$ and $x < 0$ : subtract $\pi$ . Most students only know case (1) — knowing all three prevents sign errors.

Common Mistake

The most dangerous error: applying the formula $\tan^{-1}x + \tan^{-1}y = \tan^{-1}\frac{x+y}{1-xy}$ when $xy > 1$ . This formula is only valid for $xy < 1$ . When $xy > 1$ , you need to add or subtract $\pi$ to the result. Ignoring this condition leads to answers in the wrong quadrant. Always check $xy < 1$ before using the basic form.

Question

Solution — Step by Step

Why This Works

Alternative Method — Direct substitution

Common Mistake

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