If $\cos ^{2}\left(\frac{\pi}{6}+x\right)-\sin ^{2}\left(\frac{\pi}{6}-x\right)=k \cos 2 x$ then $k=$ __________________
Given:
$\cos ^{2}\left(\frac{\pi}{6}+x\right)-\sin ^{2}\left(\frac{\pi}{6}-x\right)=K \cos 2 x$
L.H.S. is $\left(\cos \left(\frac{\pi}{6}+x\right)\right)^{2}-\left(\sin \left(\frac{\pi}{6}-x\right)\right)^{2}$
$=\left(\cos \frac{\pi}{6} \cos x-\sin \frac{\pi}{6} \sin x\right)^{2}-\left(\sin \frac{\pi}{6} \cos x-\cos \frac{\pi}{6} \sin x\right)^{2}$
using identities : -
$\cos (a+b)=\cos a \cos b-\sin a \sin b$
and $\sin (a-b)=\sin a \cos b-\cos a \sin b$
$=\left(\frac{\sqrt{3}}{2} \cos x-\frac{1}{2} \sin x\right)^{2}-\left(\frac{1}{2} \cos x-\frac{\sqrt{3}}{2} \sin x\right)^{2}$
$=\frac{3}{4} \cos ^{2} x+\frac{1}{4} \sin ^{2} x-\frac{\sqrt{3}}{2} \sin x \cos x-\left[\frac{1}{4} \cos ^{2} x+\frac{3}{4} \sin ^{2} x-\frac{\sqrt{3}}{2} \sin x \cos x\right]$
$=\frac{3}{4} \cos ^{2} x-\frac{1}{4} \cos ^{2} x+\frac{1}{4} \sin ^{2} x-\frac{3}{4} \sin ^{2} x$
$=\frac{1}{2} \cos ^{2} x-\frac{1}{2} \sin ^{2} x$
$=\frac{1}{2}\left(\cos ^{2} x-\sin ^{2} x\right)$
$=\frac{1}{2} \cos 2 x=$ R.H.S. i. e. $K \cos 2 x$
$\Rightarrow K=\frac{1}{2}$
$\therefore$ value of $K$ is $\frac{1}{2}$.
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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,