Prove that:
(i) $\sin 65^{\circ}+\cos 65^{\circ}=\sqrt{2} \cos 20^{\circ}$
(ii) $\sin 47^{\circ}+\cos 77^{\circ}=\cos 17^{\circ}$
(i) Consider LHS :
$\sin 65^{\circ}+\cos 65^{\circ}$
$=\sin 65^{\circ}+\cos \left(90^{\circ}-25^{\circ}\right)$
$=\sin 65^{\circ}+\sin 25^{\circ}$
$=2 \sin \left(\frac{65^{\circ}+25^{\circ}}{2}\right) \cos \left(\frac{65^{\circ}-25^{\circ}}{2}\right) \quad\left\{\because \sin A+\sin B=2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)\right\}$
$=2 \sin 45^{\circ} \cos 20^{\circ}$
$=2 \times \frac{1}{\sqrt{2}} \cos 20^{\circ}$
$=\sqrt{2} \cos 20^{\circ}$
$=\mathrm{RHS}$
Hence, LHS $=$ RHS.
(i) Consider LHS :
$\sin 47^{\circ}+\cos 77^{\circ}$$=\sin 47^{\circ}+\cos \left(90^{\circ}-13^{\circ}\right)$
$=\sin 47^{\circ}+\sin 13^{\circ}$
$=2 \sin \left(\frac{47^{\circ}+13^{\circ}}{2}\right) \cos \left(\frac{47^{\circ}-13^{\circ}}{2}\right) \quad\left\{\because \sin A+\sin B=2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)\right\}$
$=2 \sin 30^{\circ} \cos 17^{\circ}$
$=2 \times \frac{1}{2} \cos 17^{\circ}$
$=\cos 17^{\circ}$
$=\mathrm{RHS}$
Hence, LHS $=$ RHS.
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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,