The value of $\sin ^{2} \frac{5 \pi}{12}-\sin ^{2} \frac{\pi}{12}$ is
(a) $\frac{1}{2}$
(b) $\frac{\sqrt{3}}{2}$
(c) 1
(d) 0
(b) $\frac{\sqrt{3}}{2}$
$\frac{5 \pi}{12}=75^{\circ}, \frac{\pi}{12}=15^{\circ}$
$\sin ^{2} 75^{\circ}-\sin ^{2} 15^{\circ}$
$=\sin ^{2} 75^{\circ}-\cos ^{2} 75^{\circ} \quad\left[\sin \left(90^{\circ}-\theta\right)=\cos \theta\right]$
Now, $\sin 75^{\circ}=\sin \left(45^{\circ}+30^{\circ}\right)$
$=\sin 45^{\circ} \cos 30^{\circ}+\cos 45^{\circ} \sin 30^{\circ}$
$=\frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2}+\frac{1}{\sqrt{2}} \times \frac{1}{2}$
$=\frac{\sqrt{3}+1}{2 \sqrt{2}}$
$\cos 75^{\circ}=\cos \left(45^{\circ}+30^{\circ}\right)$
$=\cos 45^{\circ} \cos 30^{\circ}-\sin 45^{\circ} \sin 30^{\circ}$
$=\frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}} \times \frac{1}{2}$
$=\frac{\sqrt{3}-1}{2 \sqrt{2}}$
Hence,
$\sin ^{2} 75^{\circ}-\cos ^{2} 75^{\circ}=\left(\frac{\sqrt{3}+1}{2 \sqrt{2}}\right)^{2}-\left(\frac{\sqrt{3}-1}{2 \sqrt{2}}\right)^{2}$
$=\frac{3+1+2 \sqrt{3}-3-1+2 \sqrt{3}}{8}$
$=\frac{4 \sqrt{3}}{8}$
$=\frac{\sqrt{3}}{2}$
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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,