arctan(x) - arctan(y) = arctan($$\frac{x - y}{1 + xy}$$)

We will learn how to prove the property of the inverse trigonometric function arctan(x) - arctan(y) = arctan($$\frac{x - y}{1 + xy}$$) (i.e., tan$$^{-1}$$ x - tan$$^{-1}$$ y = tan$$^{-1}$$ ($$\frac{x - y}{1 + xy}$$))

Proof:

Let, tan$$^{-1}$$ x = α and tan$$^{-1}$$ y = β

From tan$$^{-1}$$ x = α we get,

x = tan α

and from tan$$^{-1}$$ y = β we get,

y = tan β

Now, tan (α - β) = ($$\frac{tan α - tan β}{1 + tan α tan β}$$)

tan (α - β) = $$\frac{x - y}{1 + xy}$$

⇒ α - β = tan$$^{-1}$$ ($$\frac{x - y}{1 + xy}$$)

⇒ tan$$^{-1}$$ x - tan$$^{-1}$$ y = tan$$^{-1}$$ ($$\frac{x - y}{1 + xy}$$)

Therefore, tan$$^{-1}$$ x - tan$$^{-1}$$ y = tan$$^{-1}$$ ($$\frac{x - y}{1 + xy}$$)

Solved examples on property of inverse circular function arctan(x) - arctan(y) = arctan($$\frac{x - y}{1 + xy}$$)

Solve the inverse trigonometric function: 3 tan$$^{-1}$$  1/2 + √3 - tan$$^{-1}$$ 1/x = tan$$^{-1}$$ 1/3

Solution:

We know that, tan 15° = tan (45° - 30°)

⇒ tan 15° = $$\frac{tan 45° - tan 30°}{1 + tan 45° tan 30°}$$

⇒ tan 15° = $$\frac{1 - \frac{1}{√3}}{1 + \frac{1}{√3}}$$

⇒ tan 15° = $$\frac{√3 - 1}{√3 + 1}$$

⇒ tan 15° = $$\frac{(√3 - 1)(√3 + 1)}{(√3 + 1)(√3 + 1)}$$

⇒ tan 15° = $$\frac{3 - 1}{4 + 2√3}$$

⇒ tan 15° = $$\frac{1}{2 + √3}$$

⇒ tan$$^{-1}$$ ($$\frac{1}{2 + √3}$$) = 15°

⇒ tan$$^{-1}$$ ($$\frac{1}{2 + √3}$$) = $$\frac{π}{12}$$

Therefore, from the given equation we get,

3 tan$$^{-1}$$ $$\frac{1}{2 + √3}$$ - tan$$^{-1}$$ $$\frac{1}{x}$$ = tan$$^{-1}$$ $$\frac{1}{3}$$

⇒ 3 · $$\frac{π}{12}$$ - tan$$^{-1}$$ $$\frac{1}{x}$$ = tan$$^{-1}$$ $$\frac{1}{3}$$

⇒ - tan$$^{-1}$$ $$\frac{1}{x}$$ = tan$$^{-1}$$ $$\frac{1}{3}$$ - $$\frac{π}{4}$$

⇒  tan$$^{-1}$$ $$\frac{1}{x}$$ = tan$$^{-1}$$ 1 - tan$$^{-1}$$ $$\frac{1}{3}$$   [Since, $$\frac{π}{4}$$ = tan$$^{-1}$$ 1]

⇒ tan$$^{-1}$$ $$\frac{1}{x}$$ = tan$$^{-1}$$ $$\frac{1 - \frac{1}{3}}{1 + 1 • \frac{1}{3}}$$

⇒ tan$$^{-1}$$ $$\frac{1}{x}$$ = tan$$^{-1}$$ ½

⇒ $$\frac{1}{x}$$  = ½

⇒ x = 2

Therefore, the required solution is x = 2.

Inverse Trigonometric Functions