In any ΔABC, prove that
$\frac{(b-c)}{a} \cos \frac{A}{2}=\sin \frac{(B-C)}{2}$
Need to prove: $\frac{(\mathrm{b}-\mathrm{c})}{\mathrm{a}} \cos \frac{\mathrm{A}}{2}=\sin \frac{(\mathrm{B}-\mathrm{C})}{2}$
We know that, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2 R$ where $R$ is the circumradius.
Therefore,
a = 2R sinA ---- (a)
Similarly, b = 2R sinB and c = 2R sinC
From Left hand side,
$=\frac{2 R \sin B-2 R \sin C}{2 R \sin A} \cos \frac{A}{2}$
$=\frac{2 \cos \left(\frac{B+C}{2}\right) \sin \left(\frac{B-C}{2}\right)}{\sin A} \cos \frac{A}{2}$
$=\frac{2 \sin \left(\frac{B-C}{2}\right) \cos \left(\frac{\pi}{2}-\frac{A}{2}\right)}{\sin A} \cos \frac{A}{2}$
$=\frac{2 \cos ^{2} \frac{A}{2} \sin \left(\frac{B-C}{2}\right)}{\sin A}$
$=\frac{\sin A \sin \left(\frac{B-C}{2}\right)}{\sin A}$
$=\sin \frac{B-C}{A}$
$=$ Right hand side. [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,