For a positive integer n, let fn(θ) = (tanθ2)(1 + sec θ

Sail E0 Webinar

Question

For a positive integer n, let

f_{n} (θ)

=

(tan \frac{θ}{2})

(1 + sec

θ

)(1 + sec 2

θ

)(1 + sec 4

θ

) .........
(1 + sec 2

^{n}

θ

). Then

Options:

A . f2(π16) = 1

B . f3(π32) = 1

C . f4(π64) = 1

D . All of above

Answer: Option D
:
D
(a,b,c)

f_{n} (θ)

=

\frac{sin (θ / 2)}{cos (θ / 2)}

[\frac{2 {cos}^{2} θ / 2}{cos θ} . \frac{2 {cos}^{2} θ}{cos 2 θ} . \frac{2 {cos}^{2} 2 θ}{cos 4 θ}]

Combine first two factors,

f_{n} (θ)

=

\frac{s i n θ}{c o s θ}

[\frac{2 {cos}^{2} θ}{cos 2 θ} . \frac{2 {cos}^{2} 2 θ}{cos 4 θ}]

Again combine first two factors,

f_{n} (θ)

= tan 2

θ

[\frac{2 {cos}^{2} 2 θ}{c o s 4 θ} . . . . . . .]

= tan (2

^{n}

θ

)

∴

f_{2} (\frac{π}{16})

= tan

\frac{4 π}{16}

= tan

(\frac{π}{4})

= 1

f_{3} (\frac{π}{32})

= tan

\frac{8 π}{32}

= tan

(\frac{π}{4})

= 1

f_{4} (\frac{π}{64})

= tan

\frac{16 π}{64}

= tan

(\frac{π}{4})

= 1

f_{5} (\frac{π}{128})

= tan

\frac{32 π}{128}

= tan

(\frac{π}{4})

= 1

Was this answer helpful ?

Next Question

Submit Solution

More Questions on This Topic :

Question 1. If α∈(0,π2) then √x2+x+tan2 α√x2+x is always greater than or equal to....

Question 2. ....

Question 3. ....

Question 4. If α+β=π2 and β+α=α, then tan α equals....

Question 5. If x1,x2,x3,......,xn are in A.P. whose common difference is a, then the value of sin α(sec x1 sec x2 + sec x2 sec....sec x2 sec x3 +....... + sec xn−1, sec xn)=

Question 6. Let n be a positive integer such that sin π2n + cos π2n = √n2. Then....

Question 7. ....

Question 8. If tan θ = √32, the sum of the infinite series 1 + 2(1 - cosθ) + 3(1 - cosθ)2 + 4(1 - cos....cosθ) + 3(1 - cosθ)2 + 4(1 - cosθ)3+........∞ is

Question 9. If tan(A−B)=1 , sec(A+B)=2√3, then the smallest positive value of B is,....

Question 10. 3[sin4(3π2−α)+sin4(3π+α)] - [sin6(π2+α)+sin6(5π−α)] =....3[sin4(3π2−α)+sin4(3π+α)] - [sin6(π2+α)+sin6(5π−α)] =

Chapter 1 - GLOBAL STEEL SCENARIO & INDIAN STEEL INDUSTRY Part 1 : (13-04-2024) INDUSTRY AND COMPANY AWARENESS (ICA)

Direction Sense Test Part 1 Reasoning (Hindi)

Chapter 1 - RMHP / OHP / OB & BP Part 1 : (14-02-2024) GPOE

Cube & Cuboid Part 1 Reasoning (Hindi)

Data Interpretation (DI) Basic Concept Reasoning (Hindi)

Counting Figures Part 1 Counting Of Straight Lines Reasoning (Hindi)

Sail E0 free webinar Webinars

Real Numbers Part 7 Class 10 Maths

Real Numbers Part 1 Class 10 Maths

Polynomials Part 1 Class 10 Maths

GPOE / GPA Combined 1 Free Mock Test SAIL JO E-0 2024

Chapter 3.1 : INTRODUCTION Chapter 3 : Sinter Plant SAIL E0 - GPOE/GPA 2024

Chapter 2.3 : COAL HANDLING PLANT - Set 1 Chapter 2 : Coke Ovens and Coal Chemicals SAIL E0 - GPOE/GPA 2024

Chapter 1.1 : INTRODUCTION Chapter 1 : Raw Material Handling Plant SAIL E0 - GPOE/GPA 2024

DSP Combined 1 Free Revision Test SAIL JO E-0