π/2
Evaluating the Sum of Inverse Tangent Functions
The problem asks us to find the value of the expression \(\tan ^{-1}(\frac{4}{5})+\tan ^{-1}(\frac{5}{4})\).
This expression involves the sum of two inverse tangent functions. Let's look closely at the arguments of the inverse tangent functions: \(\frac{4}{5}\) and \(\frac{5}{4}\). We can see that \(\frac{5}{4}\) is the reciprocal of \(\frac{4}{5}\).
We can use a standard property of inverse trigonometric functions. For any positive real number \(x\), the following identity holds:
\(\tan^{-1}(x) + \tan^{-1}(\frac{1}{x}) = \frac{\pi}{2}\)
In our given expression, we can let \(x = \frac{4}{5}\). Since \(\frac{4}{5}\) is a positive real number, the condition \(x > 0\) is satisfied.
The expression is \(\tan ^{-1}(\frac{4}{5})+\tan ^{-1}(\frac{5}{4})\).
We can write \(\tan ^{-1}(\frac{5}{4})\) as \(\tan ^{-1}(\frac{1}{4/5})\), which is in the form \(\tan^{-1}(\frac{1}{x})\) where \(x = \frac{4}{5}\).
Applying the identity \(\tan^{-1}(x) + \tan^{-1}(\frac{1}{x}) = \frac{\pi}{2}\) with \(x = \frac{4}{5}\), we get:
\(\tan ^{-1}(\frac{4}{5})+\tan ^{-1}(\frac{1}{4/5}) = \frac{\pi}{2}\)
So, \(\tan ^{-1}(\frac{4}{5})+\tan ^{-1}(\frac{5}{4}) = \frac{\pi}{2}\).
Therefore, the value of the expression is \(\frac{\pi}{2}\).
Important Questions from Inverse Trigonometric Functions
What is \(1+\sin ^2\left(\cos ^{-1}\left(\frac{3}{\sqrt{17}}\right)\right)\) equal to ?
What is 2 cot \(\left(\frac{1}{2} \cos ^{-1} \frac{\sqrt{5}}{3}\right)\) equal to ?
Consider the following statements:
1. There exists \({\rm{\theta }} \in \left( { - \frac{{\rm{\pi }}}{2},\frac{{\rm{\pi }}}{2}} \right)\) for which tan -1 (tan θ) ≠ θ
2. \({\sin ^{ - 1}}\left( {\frac{1}{3}} \right) - {\sin ^{ - 1}}\left( {\frac{1}{5}} \right) = {\sin ^{ - 1}}\left( {\frac{{2\sqrt 2 \left( {\sqrt 3 - 1} \right)}}{{15}}} \right)\)
Which of the above statements is/are correct?
Consider the following statements:
1. \({\tan ^{ - 1}}{\rm{x}} + {\tan ^{ - 1}}\left( {\frac{1}{{\rm{x}}}} \right) = {\rm{\pi }}\)
2. There exist x, y ∈ [-1, 1], where x ≠ y such that sin -1 x + cos -1 \({\rm{y}} = \frac{{\rm{\pi }}}{2}\)
Which of the above statements is/are correct?The value of \({\rm{tan}}\left( {2{{\tan }^{ - 1}}\frac{1}{5} - \frac{\pi }{4}} \right)\) is
