If in a $\Delta \mathrm{ABC}, \angle \mathrm{C}=90^{0}$, then prove that $\sin (\mathrm{A}-\mathrm{B})=\frac{\left(\mathrm{a}^{2}-\mathrm{b}^{2}\right)}{\left(\mathrm{a}^{2}+\mathrm{b}^{2}\right)}$
Given: $\angle C=90^{\circ}$
Need to prove: $\sin (A-B)=\frac{\left(a^{2}-b^{2}\right)}{\left(a^{2}+b^{2}\right)}$
Here, $\angle C=90^{\circ} ; \sin C=1$
So, it is a Right-angled triangle.
And also, $a^{2}+b^{2}=c^{2}$
Now
$\frac{a^{2}+b^{2}}{a^{2}-b^{2}} \sin (A-B)=\frac{c^{2}}{a^{2}-b^{2}} \sin (A-B)$
We know that, $\frac{\mathrm{a}}{\sin \mathrm{A}}=\frac{\mathrm{b}}{\sin \mathrm{B}}=\frac{\mathrm{c}}{\sin \mathrm{C}}=2 \mathrm{R}$ where $\mathrm{R}$ is the circumradius.
Therefore,
$=\frac{4 R^{2} \sin ^{2} C}{4 R^{2} \sin ^{2} A-4 R^{2} \sin ^{2} B} \sin (A-B)=\frac{\sin (A-B)}{\sin ^{2} A-\sin ^{2} B}[A s, \sin C=1]$
$=\frac{\sin (A-B)}{(\sin A+\sin B)(\sin A-\sin B)}=\frac{\sin (A-B)}{\left[2 \sin \frac{A+B}{2} \cos \frac{A-B}{2}\right]\left[2 \cos \frac{A+B}{2} \sin \frac{A-B}{2}\right]}$
$=\frac{\sin (A-B)}{2 \sin \frac{A+B}{2} \cos \frac{A+B}{2} .2 \sin \frac{A-B}{2} \cos \frac{A-B}{2}}=\frac{\sin (A-B)}{\sin (A+B) \sin (A-B)}$
$=\frac{1}{\sin (\pi-C)}=\frac{1}{\sin C}=1$
Therefore,
$\Rightarrow \frac{a^{2}+b^{2}}{a^{2}-b^{2}} \sin (A-B)=1$
$\Rightarrow \sin (A-B)=\frac{a^{2}-b^{2}}{a^{2}+b^{2}}$ [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,