The value of 3 (sin x – cos x)4 + 6 (sin x + cos x)2 + 4 (sin6x + cos6x) is ___________
3 (sin x – cos x)4 + 6 (sin x + cos x)2 + 4 (sin6x + cos6x)
$=3\left[(\sin x-\cos x)^{2}\right]^{2}+6\left(\sin ^{2} x+\cos ^{2} x+2 \sin x \cos x\right)+4\left[\left(\sin ^{2} x\right)^{3}+\left(\cos ^{2} x\right)^{3}\right]$
$=3\left[\sin ^{2} x+\cos ^{2} x-2 \cos x \sin x\right]^{2}+6(1+2 \sin x \cos x)$
$+4\left[\left(\sin ^{2} x+\cos _{1}^{2} x\right)^{3}-3 \sin ^{2} x \cos ^{2} x\left(\sin ^{2} x+\cos ^{2} x\right)\right]=1$
$=3[1-2 \cos x \sin x]^{2}+6+12 \sin x \cos x+4\left[1-3 \sin ^{2} x \cos ^{2} x\right]$
$=3\left[1+4 \cos ^{2} x \sin ^{2} x-4 \cos x \sin x\right]$
$+6+12 \sin x \cos x+4-12 \sin ^{2} x \cos ^{2} x$
$=3+12 \cos ^{2} x \sin ^{2} x-12 \cos x \sin x$
$+6+12 \cos x \sin x+4-12 \sin ^{2} x \cos ^{2} x$
= 13
Hence, $3(\sin x-\cos x)^{4}+6(\sin x+\cos x)^{2}+4\left(\sin ^{6} x+\cos ^{6} x\right)$ has value 13
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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,