Show that cos (2 tan-1 1/7) = sin (4 tan-1 1/3).
Taking L.H.S, we have
L.H.S. $=\cos \left(2 \tan ^{-1} \frac{1}{7}\right)$
$=\cos \left(\cos ^{-1} \frac{1-\left(\frac{1}{7}\right)^{2}}{1+\left(\frac{1}{7}\right)^{2}}\right) \quad\left(\because 2 \tan ^{-1} x=\cos ^{-1} \frac{1-x^{2}}{1+x^{2}}\right)$
$=\cos \left(\cos ^{-1} \frac{48 / 49}{50 / 49}\right)$
$=\cos \left(\cos ^{-1} \frac{24}{25}\right)=\frac{24}{25} \quad\left(\because \cos \left(\cos ^{-1} x\right)=x, x \in[-1,1]\right)$
R.H.S. $=\sin \left(4 \tan ^{-1} \frac{1}{3}\right)$
$=\sin \left(2\left(2 \tan ^{-1} \frac{1}{3}\right)\right)$
$=\sin \left(2\left(\tan ^{-1} \frac{2 \cdot \frac{1}{3}}{1-\left(\frac{1}{3}\right)^{2}}\right)\right) \quad\left(\because 2 \tan ^{-1} x \tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)\right)$
$=\sin \left(2 \tan ^{-1} \frac{2 / 3}{8 / 9}\right)$
$=\sin \left(2 \tan ^{-1} \frac{3}{4}\right)$
$=\sin \left(\sin ^{-1} \frac{2 \times \frac{3}{4}}{1+\left(\frac{3}{4}\right)^{2}}\right) \quad\left(\because 2 \tan ^{-1} x=\sin ^{-1} \frac{2 x}{1+x^{2}}\right)$
$=\sin \left(\sin ^{-1} \frac{3 / 2}{25 / 16}\right)$
$=\sin \left(\sin ^{-1} \frac{24}{25}\right)=\frac{24}{25} \quad\left(\because \sin \left(\sin ^{-1} x\right)=x, x \in[-1,1]\right)$
L.H.S. $=$ R.H.S.
Thus, L.H.S = R.H.S
– 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,