If A + B + C = π, prove that
$\cos ^{2} A+\cos ^{2} B+\cos ^{2} C=1-2 \cos A \cos B \cos C$
$=\cos ^{2} A+\cos ^{2} B+\cos ^{2} C$
Using formula,
$\frac{1+\cos 2 A}{2}=\cos ^{2} A$
$=\frac{1+\cos 2 \mathrm{~A}}{2}+\frac{1+\cos 2 \mathrm{~B}}{2}+\frac{1+\cos 2 \mathrm{C}}{2}$
$=\frac{1+\cos 2 \mathrm{~A}+1+\cos 2 \mathrm{~B}+1+\cos 2 \mathrm{C}}{2}$
$=\frac{3+\cos 2 \mathrm{~A}+\cos 2 \mathrm{~B}+\cos 2 \mathrm{C}}{2}$
Using,
$\cos A+\cos B=2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)$
$=\frac{3+\cos 2 \mathrm{~A}+2 \cos \left(\frac{2 \mathrm{~B}+2 \mathrm{C}}{2}\right) \cos \left(\frac{2 \mathrm{~B}-2 \mathrm{C}}{2}\right)}{2}$
$=\frac{3+\cos 2 \mathrm{~A}+2 \cos (\mathrm{B}+\mathrm{C}) \cos (\mathrm{B}-\mathrm{C})}{2}$
Using, since $A+B+C=\pi$
$\rightarrow \mathrm{B}+\mathrm{C}=180-\mathrm{A}$
And, $\cos (\pi-A)=-\cos A$
$=\frac{3+\cos 2 A+2 \cos (\pi-A) \cos (B-C)}{2}$
$=\frac{3+\cos 2 A-2 \cos (A) \cos (B-C)}{2}$
Using $\cos 2 A=2 \cos ^{2} A-1$
$=\frac{3+2 \cos ^{2} A-1-2 \cos (A) \cos (B-C)}{2}$
$=\frac{2+2 \cos ^{2} A-2 \cos (A) \cos (B-C)}{2}$
$=1+\cos ^{2} A-\cos A \cos (B-C)$
$=1+\cos A\{\cos A-\cos (B-C)\}$
Using,
$\cos A-\cos B=2 \sin \left(\frac{A+B}{2}\right) \sin \left(\frac{B-A}{2}\right)$
$=1+\cos A\left(2 \sin \left(\frac{A+B-C}{2}\right) \sin \left(\frac{B-C-A}{2}\right)\right)$
Since,$A+B+C=\pi$
$=1+\cos A\left(2 \sin \left(\frac{\pi-C-C}{2}\right) \sin \left(\frac{B-(\pi-B)}{2}\right)\right)$
$=1+\cos A\left(2 \cos C \sin \left(\frac{B}{2}-\frac{\pi}{2}\right)\right)$
$=1-2 \cos A \cos C \cos C$
$=$ R.H.S
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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,