Prove that:
$\cot ^{-1}\left(\sqrt{1+x^{2}}-x\right)=\frac{\pi}{2}-\frac{1}{2} \cot ^{-1} x$
To Prove: $\cot ^{-1}\left(\sqrt{1+x^{2}}-x\right)=\frac{\pi}{2}-\frac{1}{2} \cot ^{-1} x$
Formula Used:
1) $\tan \left(\frac{\pi}{4}+\mathrm{A}\right)=\frac{1+\tan \mathrm{A}}{1-\tan \mathrm{A}}$
2) $\operatorname{cosec}^{2} A=1+\cot ^{2} A$
3) $1-\cos A=2 \sin ^{2}\left(\frac{A}{2}\right)$
4) $\sin A=2 \sin \left(\frac{A}{2}\right) \cos \left(\frac{A}{2}\right)$
Proof:
$\mathrm{LHS}=\cot ^{-1}\left(\sqrt{1+\mathrm{x}^{2}}-\mathrm{x}\right)$
Let $x=\cot A$
$\mathrm{LHS}=\cot ^{-1}\left(\sqrt{1+\cot ^{2} \mathrm{~A}}-\cot \mathrm{A}\right)$
$=\cot ^{-1}(\operatorname{cosec} A-\cot A)$
$=\cot ^{-1}\left(\frac{1-\cos \mathrm{A}}{\sin \mathrm{A}}\right)$
$=\cot ^{-1}\left(\frac{2 \sin ^{2}\left(\frac{\mathrm{A}}{2}\right)}{2 \sin \left(\frac{\mathrm{A}}{2}\right) \cos \left(\frac{\mathrm{A}}{2}\right)}\right)$
$=\cot ^{-1}\left(\tan \left(\frac{\mathrm{A}}{2}\right)\right)$
$=\frac{\pi}{2}-\tan ^{-1}\left(\tan \left(\frac{\mathrm{A}}{2}\right)\right)$
$=\frac{\pi}{2}-\frac{\mathrm{A}}{2}$
From $(2), A=\cot ^{-1} x$
$\frac{\pi}{2}-\frac{A}{2}=\frac{\pi}{2}-\frac{1}{2} \cot ^{-1} x$
$=\mathrm{RHS}$
Therefore, LHS = RHS
Hence proved.
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,