If A + B + C = π, prove that
$\frac{\cos A}{\sin B \sin C}+\frac{\cos B}{\sin C \sin A}+\frac{\cos C}{\sin A \sin B}=2$
$=\frac{\cos A}{\sin B \sin C}+\frac{\cos B}{\sin C \sin A}+\frac{\cos C}{\sin A \sin B}$
Taking L.C.M
$=\frac{\cos A \sin A+\cos B \sin B+\cos C \sin C}{\sin B \sin C \sin A}$
Multiplying and divide the above equation by 2, we get
$=\frac{2 \cos A \sin A+2 \cos B \sin B+2 \cos C \sin C}{2 \sin B \sin C \sin A}$
Since , sin2A = 2sinAcosA
$=\frac{\sin 2 A+\sin 2 B+\sin 2 C}{2 \sin B \sin C \sin A}$
Now,
$=\sin 2 A+\sin 2 B+\sin 2 C$
$=2 \sin A \cos A+2 \sin (B+C) \cos (B-C)$
since A + B + C = π
$\rightarrow \mathrm{B}+\mathrm{A}=180-\mathrm{C}$
$=2 \sin A \cos A+2 \sin (\pi-A) \cos (B-C)$
$=2 \sin A \cos A+2 \sin A \cos (B-C)$
$=2 \sin A\{\cos A+\cos (B-C)\}$
$($ but $\cos A=\cos \{180-(B+C)\}=-\cos (B+C)$
And now using $\cos A-\cos B=2 \sin \left(\frac{A+B}{2}\right) \sin \left(\frac{-A+B}{2}\right)$
$=2 \sin A\{2 \sin B \sin C\}$
$=4 \sin A \sin B \sin C$
Putting the above value in the equation, we get
$=\frac{4 \sin A \sin B \sin C}{2 \sin B \sin C \sin A}$
= 2
= 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,