Show that: $3(\sin x-\cos x)^{4}+6(\sin x+\cos )^{2}+4\left(\sin ^{6} x+\cos ^{6} x\right)=13$
$\mathrm{LHS}=3(\sin x-\cos x)^{4}+6(\sin x+\cos x)^{2}+4\left(\sin ^{6} x+\cos ^{6} x\right)$
$=3\left\{(\sin x-\cos x)^{2}\right\}^{2}+6\left(\sin ^{2} x+\cos ^{2} x+2 \sin x \cos x\right)+4\left\{\left(\sin ^{2} x\right)^{3}+\left(\cos ^{2} x\right)^{3}\right\}$
$=3\left(\sin ^{2} x+\cos ^{2} x-2 \sin x \cos x\right)^{2}+6(1+\sin 2 x)+4\left(\sin ^{2} x+\cos ^{2} x\right)\left(\sin ^{4} x-\sin ^{2} x \cos ^{2} x+\cos ^{4} x\right)$
$=3(1-\sin 2 x)^{2}+6+6 \sin 2 x+4\left(\sin ^{4} x-\sin ^{2} x \cos ^{2} x+\cos ^{4} x\right)$
$=3(1-\sin 2 x)^{2}+6+6 \sin 2 x+4\left\{\left(\sin ^{2} x\right)^{2}+\left(\cos ^{2} x\right)^{2}\right\}-12 \sin ^{2} x \cos ^{2} x$
$=3(1-\sin 2 x)^{2}+6+6 s \operatorname{in} 2 x+4\left\{\left(\sin ^{2} x+\cos ^{2} x\right)^{2}-2 \sin ^{2} x \cos ^{2} x\right\}-12 \sin ^{2} x \cos ^{2} x$
$=3(1-\sin 2 x)^{2}+6+6 \sin 2 x+4\left\{1-2 \sin ^{2} x \cos ^{2} x\right\}-12 \sin ^{2} x \cos ^{2} x$
$=3\left(1-2 \sin 2 x+\sin ^{2} 2 x\right)+6+6 \sin 2 x+4-20 \sin ^{2} x \cos ^{2} x$
$=-6 \sin 2 x+3 \sin ^{2} 2 x+13+6 \sin 2 x-5(2 \sin x \cos x)^{2}$
$=3 \sin ^{2} 2 x+13-5 \sin ^{2} 2 x$
$=13-2 \sin ^{2} 2 x=\mathrm{RHS}$
Hence 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,