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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