Question:
Prove that: $\cot ^{2} x-\tan ^{2} x=4 \cot 2 x \operatorname{cosec} 2 x$
Solution:
$\mathrm{LHS}=\cot ^{2} x-\tan ^{2} x$
$=\frac{\cos ^{2} x}{\sin ^{2} x}-\frac{\sin ^{2} x}{\cos ^{2} x}$
$=\frac{\left(\cos ^{2} x\right)^{2}-\left(\sin ^{2} x\right)^{2}}{\sin ^{2} x \cos ^{2} x}$
$=\frac{\left(\cos ^{2} x+\sin ^{2} x\right)\left(\cos ^{2} x-\sin ^{2} x\right)}{\sin ^{2} x \cos ^{2} x}$
$=\frac{1 \times(\cos 2 x)}{\sin ^{2} x \cos ^{2} x} \quad\left(\because \cos ^{2} x-\sin ^{2} x=\cos 2 x\right)$
$=\frac{4 \cos 2 x}{4 \sin ^{2} x \cos ^{2} x}$
$=\frac{4(\cos 2 x)}{(\sin 2 x)^{2}}$
$=\frac{4(\cos 2 x)}{(\sin 2 x)} \times \frac{1}{(\sin 2 x)}$
$=4 \cot 2 x \operatorname{cosec} 2 x=\mathrm{RHS}$
Hence proved.
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- 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
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- Dual Nature of Radiation and Matter
- Atoms
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- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,
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