arctan(x) + arccot(x) = $\frac{π}{2}$

We will learn how to prove the property of the inverse trigonometric function arctan(x) + arccot(x) = $\frac{π}{2}$ (i.e., tan$^{-1}$ x + cot$^{-1}$ x = $\frac{π}{2}$).

Proof: Let, tan$^{-1}$ x = θ              

Therefore, x = tan θ

x = cot ($\frac{π}{2}$ - θ), [Since, cot ($\frac{π}{2}$ - θ) = tan θ]

⇒ cot$^{-1}$ x = $\frac{π}{2}$ - θ

⇒ cot$^{-1}$ x= $\frac{π}{2}$ - tan$^{-1}$ x, [Since, θ = tan$^{-1}$ x]

⇒ cot$^{-1}$ x + tan$^{-1}$ x = $\frac{π}{2}$

⇒ tan$^{-1}$ x + cot$^{-1}$ x = $\frac{π}{2}$

Therefore, tan$^{-1}$ x + cot$^{-1}$ x = $\frac{π}{2}$.             Proved.

Solved examples on property of inverse circular function tan$^{-1}$ x + cot$^{-1}$ x = $\frac{π}{2}$

Prove that, tan$^{-1}$ 4/3 + tan$^{-1}$ 12/5 = π - tan$^{-1}$ $\frac{56}{33}$.

Solution:  

We know that tan$^{-1}$ x + cot$^{-1}$ x = $\frac{π}{2}$

⇒ tan$^{-1}$ x = $\frac{π}{2}$ - cot$^{-1}$ x

⇒ tan$^{-1}$ $\frac{4}{3}$  = $\frac{π}{2}$ - cot$^{-1}$ $\frac{4}{3}$

and

tan$^{-1}$ $\frac{12}{5}$ = $\frac{π}{2}$ - cot$^{-1}$ $\frac{12}{5}$

Now, L. H. S. = tan$^{-1}$ $\frac{4}{3}$ + tan$^{-1}$ $\frac{12}{5}$

= $\frac{π}{2}$ - cot$^{-1}$ $\frac{4}{3}$ + $\frac{π}{2}$ - cot$^{-1}$ $\frac{12}{5}$, [Since, tan$^{-1}$ $\frac{4}{3}$ = $\frac{π}{2}$ - cot$^{-1}$ $\frac{4}{3}$ and tan$^{-1}$ $\frac{12}{5}$ = $\frac{π}{2}$ - cot$^{-1}$ $\frac{12}{5}$]

= π - (cot$^{-1}$ $\frac{4}{3}$ + cot$^{-1}$ $\frac{12}{5}$)

= π - (tan$^{-1}$ $\frac{3}{4}$ + tan$^{-1}$ $\frac{5}{12}$)

= π – tan$^{-1}$ $\frac{\frac{3}{4} + \frac{5}{12}}{1 – \frac{3}{4} · \frac{5}{12}}$

= π – tan$^{-1}$ ($\frac{14}{12}$ x $\frac{48}{33}$)

= π – tan$^{-1}$ $\frac{56}{33}$ = R. H. S.       Proved.

● Inverse Trigonometric Functions

• General and Principal Values of sin$^{-1}$ x
• General and Principal Values of cos$^{-1}$ x
• General and Principal Values of tan$^{-1}$ x
• General and Principal Values of csc$^{-1}$ x
• General and Principal Values of sec$^{-1}$ x
• General and Principal Values of cot$^{-1}$ x
• Principal Values of Inverse Trigonometric Functions
• General Values of Inverse Trigonometric Functions
  
• arcsin(x) + arccos(x) = $\frac{π}{2}$
• arctan(x) + arccot(x) = $\frac{π}{2}$
• arctan(x) + arctan(y) = arctan($\frac{x + y}{1 - xy}$)
• arctan(x) - arctan(y) = arctan($\frac{x - y}{1 + xy}$)
• arctan(x) + arctan(y) + arctan(z)= arctan$\frac{x + y + z – xyz}{1 – xy – yz – zx}$
• arccot(x) + arccot(y) = arccot($\frac{xy - 1}{y + x}$)
• arccot(x) - arccot(y) = arccot($\frac{xy + 1}{y - x}$)
• arcsin(x) + arcsin(y) = arcsin(x $\sqrt{1 - y^{2}}$ + y$\sqrt{1 - x^{2}}$)
• arcsin (x) - arcsin(y) = arcsin (x $\sqrt{1 - y^{2}}$ - y$\sqrt{1 - x^{2}}$)
• arccos (x) + arccos(y) = arccos(xy - $\sqrt{1 - x^{2}}$$\sqrt{1 - y^{2}}$)
• arccos(x) - arccos(y) = arccos(xy + $\sqrt{1 - x^{2}}$$\sqrt{1 - y^{2}}$)
• 2 arcsin(x) = arcsin(2x$\sqrt{1 - x^{2}}$) 
• 2 arccos(x) = arccos(2x$^{2}$ - 1)
• 2 arctan(x) = arctan($\frac{2x}{1 - x^{2}}$) = arcsin($\frac{2x}{1 + x^{2}}$) = arccos($\frac{1 - x^{2}}{1 + x^{2}}$)
• 3 arcsin(x) = arcsin(3x - 4x$^{3}$)
• 3 arccos(x) = arccos(4x$^{3}$ - 3x)
• 3 arctan(x) = arctan($\frac{3x - x^{3}}{1 - 3 x^{2}}$)
• Inverse Trigonometric Function Formula
• Principal Values of Inverse Trigonometric Functions
• Problems on Inverse Trigonometric Function

11 and 12 Grade Math

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