If θ1, θ2, θ3, …, θn are in A.P., whose common difference is d, show that Sec θ1 sec θ2 + sec θ2 sec θ3 + … + sec θn–1 sec θn
$=\frac{\tan \theta_{\mathrm{n}}-\tan \theta_{1}}{\sin \mathrm{d}}$
Given $\theta_{1}, \theta_{2}, \theta_{3}, \ldots, \theta_{n}$ are in A.P., and common difference is $d$, Now we have to prove that
$\sec \theta_{1} \sec \theta_{2}+\sec \theta_{2} \sec \theta_{3}+\ldots+\sec \theta_{n-1} \sec \theta_{n}=\frac{\tan \theta_{n}-\tan \theta_{1}}{\sin d}$
On cross multiplication we get
$\Rightarrow \sin d\left(\sec \theta_{1} \sec \theta_{2}+\sec \theta_{2} \sec \theta_{3}+\ldots+\sec \theta_{n-1} \sec \theta_{n}\right)=\tan \theta_{n}-\tan \theta_{1}$
We know sec $x=1 / \cos x$ using this formula we get
$\Rightarrow \frac{\sin d}{\cos \theta_{1} \cos \theta_{2}}+\frac{\text { sind }}{\cos \theta_{2} \cos \theta_{3}}+\cdots+\frac{\operatorname{sind}}{\cos \theta_{n-1} \cos \theta_{n}}=\tan \theta_{n}-\tan \theta_{1}$
Consider LHS
$\Rightarrow \mathrm{LHS}=\frac{\text { sind }}{\cos \theta_{1} \cos \theta_{2}}+\frac{\text { sind }}{\cos \theta_{2} \cos \theta_{3}}+\cdots+\frac{\text { sind }}{\cos \theta_{\mathrm{n}-1} \cos \theta_{\mathrm{n}}}$
Now we have to find value of $d$ in terms of $\theta$ so that further simplification can be made
As $\theta_{1}, \theta_{2}, \theta_{3}, \ldots, \theta_{n}$ are in AP having common difference as $d$
Hence
$\theta_{2}-\theta_{1}=d, \theta_{3}-\theta_{2}=d, \ldots, \theta_{n}-\theta_{n-1}=d$
Take sin on both sides
$\sin \left(\theta_{2}-\theta_{1}\right)=\sin d, \sin \left(\theta_{3}-\theta_{2}\right)=\sin d, \ldots, \sin \left(\theta_{n}-\theta_{n-1}\right)=\sin d$
Substitute appropriate value of sin d for each term in LHS
$\Rightarrow \mathrm{LHS}=\frac{\sin \left(\theta_{2}-\theta_{1}\right)}{\cos \theta_{1} \cos \theta_{2}}+\frac{\sin \left(\theta_{3}-\theta_{2}\right)}{\cos \theta_{2} \cos \theta_{3}}+\cdots+\frac{\sin \left(\theta_{\mathrm{n}}-\theta_{\mathrm{n}-1}\right)}{\cos \theta_{\mathrm{n}-1} \cos \theta_{\mathrm{n}}}$
We know that $\sin (a-b)=\sin a \cos b-\cos a \sin b$
Using this formula we get
$\Rightarrow \mathrm{LHS}=\frac{\sin \theta_{2} \cos \theta_{1}-\cos \theta_{2} \sin \theta_{1}}{\cos \theta_{1} \cos \theta_{2}}+\frac{\sin \theta_{3} \cos \theta_{2}-\cos \theta_{3} \sin \theta_{2}}{\cos \theta_{2} \cos \theta_{3}}+\cdots$
On simplifying we get
$=\frac{\sin \theta_{2} \cos \theta_{1}}{\cos \theta_{1} \cos \theta_{2}}-\frac{\cos \theta_{2} \sin \theta_{1}}{\cos \theta_{1} \cos \theta_{2}}+\frac{\sin \theta_{3} \cos \theta_{2}}{\cos \theta_{2} \cos \theta_{3}}-\frac{\cos \theta_{3} \sin \theta_{2}}{\cos \theta_{2} \cos \theta_{3}}+\cdots+\frac{\sin \theta_{n} \cos \theta_{n-1}}{\cos \theta_{n-1} \cos \theta_{n}}$ $-\frac{\cos \theta_{n} \sin \theta_{n-1}}{\cos \theta_{n-1} \cos \theta_{n}}$
We know that $\sin x / \cos x=\tan x$
$=\tan \theta_{2}-\tan \theta_{1}+\tan \theta_{3}-\tan \theta_{2}+\ldots+\tan \theta_{n}-\tan \theta_{n-1}$
$=-\tan \theta_{1}+\tan \theta_{n}$
$=\tan \theta_{n}-\tan \theta_{1}$
$\Rightarrow \mathrm{LHS}=\mathrm{RHS}$
Hence proved
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- 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,