Topics

General Knowledge
General Science
General English
Aptitude
General Computer Science
General Intellingence and Reasoning
Current Affairs
Exams
Elementary Mathematics
English Literature

If \( \Large \cos \theta = \frac{1}{2} \left(x+\frac{1}{x}\right),\ then\ \frac{1}{2} \left(x^{2} + \frac{1}{x^{2}}\right) \) is equal to:

A) \( \Large \sin 2 \theta \)
B) \( \Large \sec 2 \theta \)
C) \( \Large \tan 2 \theta \)
D) \( \Large \cos 2 \theta \)

Correct Answer:




D) \( \Large \cos 2 \theta \)

Description for Correct answer:

Given that \( \Large \cos \theta = \frac{1}{2} \left(x+\frac{1}{x}\right) => x+\frac{1}{x}=2 \cos \theta \)

We know that \( \Large x^{2}+\frac{1}{x^{2}} = \left(x+\frac{1}{x}\right)^{2}-2 \)

= \( \Large \left(2 \cos \theta \right)^{2}-2=4 \cos^{2} \theta -2 \)

= \( \Large 2 \cos 2 \theta \) .. From (i)

Therefore, \( \Large \frac{1}{2} \left(x^{2}+\frac{1}{x^{2}}\right)=\frac{1}{2} \times 2 \cos \theta = \cos 2 \theta \)

Part of solved Trigonometric ratio questions and answers : >> Elementary Mathematics >> Trigonometric ratio

Comments

Similar Questions

1). The value of x for the maximum value \( \Large \sqrt{3} \cos x + \sin x \) is:

A). \( \Large 30 ^{\circ} \)

B). \( \Large 45 ^{\circ} \)

C). \( \Large 60 ^{\circ} \)

D). \( \Large 90 ^{\circ} \)

2). The equation \( \Large \left(a+b\right)^{2}=4ab \sin^{2} \theta \) is possible only when

A). \( \Large a = b \)

B). \( \Large 2a = b \)

C). \( \Large a = 2b \)

D). none of these

3). The value of the expression \( \Large 1 - \frac{\sin^{2} y}{1+\cos y}+\frac{1 + \cos y}{\sin y}-\frac{\sin y}{1- \cos y} \) is equal to;

A). 0

B). 1

C). \( \Large \sin y \)

D). \( \Large \cos y \)

4). The circular wire of diameter 10 cm is cut and placed along the circumference of a circle of diameter 1m. The angle subtended by the write at the centre of the circle is equal to:

A). \( \Large \frac{ \pi }{4}rad \)

B). \( \Large \frac{ \pi }{3}rad \)

C). \( \Large \frac{ \pi }{5}rad \)

D). \( \Large \frac{ \pi }{10}rad \)

5). The greatest and least value of \( \Large \sin x \cos\ x \) are:

A). 1, -1

B). \( \Large \frac{1}{2},\ -\frac{1}{2} \)

C). \( \Large \frac{1}{4},\ -\frac{1}{4} \)

D). 2, -2

6). If \( \Large A = \sin^{2} \theta + \cos^{4} \theta \), then for all real values of \( \Large \theta \)

A). \( \Large 1 \le A \le 2 \)

B). \( \Large \frac{3}{4} \le A \le 1 \)

C). \( \Large \frac{13}{16} \le A \le 1 \)

D). \( \Large \frac{3}{4} \le A \le \frac{13}{16} \)

7). \( \Large \tan \frac{2 \pi }{5} - \tan \frac{ \pi }{15} - \sqrt{3} \tan \frac{2 \pi }{5} \tan \frac{ \pi }{15} \) is equal to:

A). \( \Large -\sqrt{3} \)

B). \( \Large \frac{1}{\sqrt{3}} \)

C). \( \Large 1 \)

D). \( \Large \sqrt{3} \)

8). If \( \Large A = 130 ^{\circ} and\ x=\sin A + \cos A \), then:

A). \( \Large x > 0 \)

B). \( \Large x < 0 \)

C). \( \Large x=0 \)

D). \( \Large x \le 0 \)

9). If \( \Large A+B+C= \pi \ and\ \cos A = B \cos C,\ then\ \tan B \tan C \) is equal to:

A). \( \Large \frac{1}{2} \)

B). 2

C). 1

D). \( \Large -\frac{1}{2} \)

10). The period of \( \Large \sin^{2} \theta \) is

A). \( \Large \pi ^{2} \)

B). \( \Large \pi \)

C). \( \Large 2 \pi \)

D). \( \Large \frac{ \pi }{2} \)