If A + B + C = π, prove that
$\sin (B+C-A)+\sin (C+A-B)-\sin (A+B-C)=4 \cos A \cos B \sin C$
$=\sin (B+C-A)+\sin (C+A-B)-\sin (A+B-C)$
Using,
$\sin A+\sin B=2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)$
$=2 \sin C \cos (B-A)-\sin (A+B-C)$
Since $A+B+C=\pi$
$\rightarrow \mathrm{B}+\mathrm{A}=180-\mathrm{C}$
$=2 \sin C \cos (B-A)-\sin (\pi-C-C)$
$=2 \sin C \cos (B-A)-\sin 2 C$
Since , sin2A = 2sinAcosA,
$=2 \sin C \cos (B-A)-2 \sin C \cos C$
$=2 \sin C\{\cos (B-A)-\cos C\}$
Using ,
$\cos A-\cos B=2 \sin \left(\frac{A+B}{2}\right) \sin \left(\frac{B-A}{2}\right)$
$=2 \sin C\left\{2 \sin \left(\frac{B-A+C}{2}\right) \sin \left(\frac{C-B+A}{2}\right)\right\}$
$=2 \sin C\left\{2 \sin \left(\frac{\pi-A-A}{2}\right) \sin \left(\frac{\pi-B-B}{2}\right)\right\}$
= 4cosAcosBsinC
= 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,