KCET · Maths · Inverse Trigonometric Functions
The value of \(\tan ^{-1}\left(\frac{x}{y}\right)-\tan ^{-1}\left(\frac{x-y}{x+y}\right)\) (where, \(x, y>0\) ) is
- A \(\frac{\pi}{4}\)
- B \(-\frac{\pi}{4}\)
- C \(\frac{\pi}{2}\)
- D \(-\frac{\pi}{2}\)
Answer & Solution
Correct Answer
(A) \(\frac{\pi}{4}\)
Step-by-step Solution
Detailed explanation
\(\tan ^{-1}\left(\frac{x}{y}\right)-\tan ^{-1}\left(\frac{x-y}{x+y}\right), \quad(\because x, y>0)\)
\(\quad=\tan ^{-1}\left\{\frac{\frac{x}{y}-\left(\frac{x-y}{x+y}\right)}{1+\frac{x(x-y)}{y(x+y)}}\right\}\)
\(\begin{aligned} & {\left[\because \tan ^{-1} A-\tan ^{-1} B=\tan ^{-1}\left\{\frac{A-B}{1+A B}\right\}\right] } \\=& \tan ^{-1}\left(\frac{x^{2}+x y-x y+y^{2}}{x y+y^{2}+x^{2}-x y}\right) \\=& \tan ^{-1}\left(\frac{x^{2}+y^{2}}{x^{2}+y^{2}}\right) \\=& \tan ^{-1}(1)=\tan ^{-1}\left(\tan \frac{\pi}{4}\right)=\frac{\pi}{4} \end{aligned}\)
\(\quad=\tan ^{-1}\left\{\frac{\frac{x}{y}-\left(\frac{x-y}{x+y}\right)}{1+\frac{x(x-y)}{y(x+y)}}\right\}\)
\(\begin{aligned} & {\left[\because \tan ^{-1} A-\tan ^{-1} B=\tan ^{-1}\left\{\frac{A-B}{1+A B}\right\}\right] } \\=& \tan ^{-1}\left(\frac{x^{2}+x y-x y+y^{2}}{x y+y^{2}+x^{2}-x y}\right) \\=& \tan ^{-1}\left(\frac{x^{2}+y^{2}}{x^{2}+y^{2}}\right) \\=& \tan ^{-1}(1)=\tan ^{-1}\left(\tan \frac{\pi}{4}\right)=\frac{\pi}{4} \end{aligned}\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- Find the co-ordinates of the foot of the perpendicular drawn from the origin to the plane \( 5 y+8 \)
\( =0 \) :KCET 2016 Hard - If \(A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]\), then \(\mathrm{A}^{-1}\) is equal toKCET 2008 Easy
- A die is thrown 10 times, the probability that an odd number will come up at least one time isKCET 2020 Medium
- Inverse of a diagonal non-singular matrix isKCET 2012 Medium
- The sum of all positive divisors of 242 except 1 and itself isKCET 2011 Easy
- The function \(f(x)=\frac{x}{3}+\frac{3}{x}\) decreases in the intervalKCET 2013 Easy
More PYQs from KCET
- Generally, the first ionisation energy increases along a period. But there are some exceptions. The one which is not an exception isKCET 2011 Easy
- If \( \vec{A}=2 \hat{i}+3 \hat{j}+8 \hat{k} \) is perpendicular to \( \vec{B}=4 \hat{j}-4 \hat{i}+\alpha \hat{k} \), then the value of ' \( \alpha \) ' isKCET 2017 Easy
- The metal nitrate that liberates \( \mathrm{NO}_{2} \) on heatingKCET 2019 Easy
- Which examples of carbohydrates exhibit \(\alpha\)-link, ( \(\alpha\)-glycosidic link) in their structure?KCET 2025 Easy
- If \( y=(1+x)\left(1+x^{2}\right)\left(1+x^{4}\right) \), then \( \frac{d y}{d x} \) at \( x=1 \) isKCET 2014 Easy
- Left auricle receives pure blood from theKCET 2013 Medium