MHT CET · Maths · Inverse Trigonometric Functions
If \(\cot ^{-1}(\sqrt{\cos \alpha})-\tan ^{-1}(\sqrt{\cos \alpha})=x\), then the value of \(\sin x\) is
- A \(\cot ^2\left(\frac{\alpha}{2}\right)\)
- B \(\tan ^2\left(\frac{\alpha}{2}\right)\)
- C \(\tan \alpha\)
- D \(\cot \left(\frac{\alpha}{2}\right)\)
Answer & Solution
Correct Answer
(B) \(\tan ^2\left(\frac{\alpha}{2}\right)\)
Step-by-step Solution
Detailed explanation
\(\begin{gathered}\cot ^{-1}(\sqrt{\cos \alpha})-\tan ^{-1}(\sqrt{\cos \alpha})=x \\ \Rightarrow \tan ^{-1}\left[\frac{1}{\sqrt{\cos \alpha}}\right]-\tan ^{-1}[\sqrt{\cos \alpha}]=x \\ \Rightarrow \tan ^{-1}\left[\frac{\frac{1}{\sqrt{\cos \alpha}}-\sqrt{\cos \alpha}}{1+\frac{\sqrt{\cos \alpha}}{\sqrt{\cos \alpha}}}\right]=x \\ \Rightarrow \tan x=\frac{1-\cos \alpha}{2 \sqrt{\cos \alpha}} \\ \therefore \quad \sin x=\frac{1-\cos \alpha}{1+\cos \alpha}=\frac{2 \sin ^2 \frac{\alpha}{2}}{2 \cos ^2 \frac{\alpha}{2}} \\ \quad=\tan ^2\left(\frac{\alpha}{2}\right)\end{gathered}\)
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
- If is centroid of triangle where, and are vertices of a triangle then values of are respectively ________MHT CET 2019 Easy
- If \(1-\cos \theta=\sin \theta \cdot \sin \frac{\theta}{2}\), then the value of \(\theta\) isMHT CET 2025 Medium
- The inverse of the proposition \((p \wedge-q) \Rightarrow r\) isMHT CET 2011 Easy
- In a triangle \(A B C\), with usual notations \(a=2, b=3, c=5\), then \(\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}=\)MHT CET 2021 Easy
- \(\int \cos ^{\frac{-3}{7}} x \cdot \sin ^{\frac{-11}{7}} x d x=\)MHT CET 2024 Hard
- If the point (\(1, \alpha, \beta\)) lies on the line of the shortest distance between the lines \(\frac{x+2}{-3}=\frac{y-2}{4}=\frac{z-5}{2}\) and \(\frac{x+2}{-1}=\frac{y+6}{2}, z=1\), then \(\alpha+\beta=\)MHT CET 2025 Hard
More PYQs from MHT CET
- An average induced emf of \(0.5 \mathrm{~V}\) appears in a coil when the current in it is changed from \(10 \mathrm{~A}\) in one direction to \(10 \mathrm{~A}\) in opposite direction in 0.8 second. The self-inductance of the coil isMHT CET 2022 Easy
- If \(\int \frac{2 x^2+3}{\left(x^2-1\right)\left(x^2-4\right)} \mathrm{d} x=\log \left[\left(\frac{x-2}{x+2}\right)^a \cdot\left(\frac{x+1}{x-1}\right)^b\right]+c\),
(where c is the constant of integration) then the value of \(a+\mathrm{b}\) is equal toMHT CET 2025 Medium - If \(\sin \left(\cot ^{-1}(x+1)\right)=\cos \left(\tan ^{-1} x\right)\) then considering positive square roots, \(x\) has the value \(\qquad\)MHT CET 2024 Medium
- A particle of mass ' \(m\) ' is performing uniform circular motion along a circular path of radius ' \(r\) '. Its angular momentum about the axis passing through the centre and perpendicular to the plane is ' \(L\) '. The kinetic energy of the particle isMHT CET 2024 Easy
- If \(\mathrm{f}(x)=x^3+\mathrm{b} x^2+\mathrm{c} x+\mathrm{d}\) and \(0 \lt \mathrm{b}^2 \lt \mathrm{c}\), then in \((-\infty, \infty)\)MHT CET 2024 Medium
- In the ecological hierarchy the basic unit is _________.MHT CET 2021 Medium