Topics

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

Consider the following equations

\( \Large cosec^{2}x+sec^{2}x=cosec^{2}x \ sec^{2}x \)

\( \Large sec \ x+tan^{2}x=sec \ x \ tan^{2}x \)

\( \Large cosec \ x+tan^{2}x=cot^{2}x+sec \ x \)

Which of the above equations are correct?

A) (i) and (ii)
B) (ii) and (iii)
C) (i) and (iii)
D) (i), (ii) and (iii)

Correct Answer:



C) (i) and (iii)

Description for Correct answer:

(i) LHS = \( \Large cosec^{2}x+sec^{2}x \)

= \( \Large \frac{1}{sin^{2}x}+\frac{1}{cos^{2}x}=\frac{cos^{2}x+sin^{2}x}{sin^{2}x \ cos^{2}x} \)

= \( \Large \frac{1}{sin^{2}x \ cos^{2}x}=\frac{1}{sin^{2}x}.\frac{1}{cos^{2}x} \)

= \( \Large cosec^{2}x \ sec^{2}x=RHS \)

(ii)\( \Large LHS = sec^{2}x + tan^{2}x \)

\( \Large \frac{1}{cos^{2}x} + \frac{sin^{2}x}{cos^{2}x} \)

\( \Large \frac{1 + sin^{2}}{cos^{2}x} \ne sec^{2}x tan^{2}x = RHS\)

(iii) LHS = \( \Large cosec^{2}x+tan^{2}x \)

= \( \Large 1+cot^{2}x+sec^{2}x-1 \)

\( \Large cosec^{2}x=1+cot^{2}x \)

=> \( \Large sec^{2}x=1+tan^{2}x \)

= \( \Large cot^{2}x+sec^{2}x \)

so, Eqs. (i) and (iii) are correct.

Part of solved Trigonometry questions and answers : >> Elementary Mathematics >> Trigonometry

Comments

Similar Questions

1). If \( \Large \frac{cos \ x}{1+cosec \ x}+\frac{cosx}{cosec\ x-1}=2 \) ,which one of the following is one of the value of x ?

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

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

C). \( \Large \frac{ \pi }{4} \)

D). \( \Large \frac{ \pi }{6} \)

2). \( \Large \frac{cos \theta }{1-sin \theta }-\frac{cos \theta }{1+sin \theta }=2 \),is satisfied by which one of the following values of \( \Large \theta \)?

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

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

C). \( \Large \frac{ \pi }{4} \)

D). \( \Large \frac{ \pi }{6} \)

3). What is \( \Large \sqrt{\frac{1+sin \theta }{1-sin \theta }} \) equal to?

A). \( \Large sec \theta -tan \theta \)

B). \( \Large sec \theta +tan \theta \)

C). \( \Large cosec \theta +cot \theta \)

D). \( \Large cosec \theta -cot \theta \)

4). If \( \Large 7cos^{2} \theta+3sin^{2} \theta =4 \ and \ 0 \ < \ \theta \ < \frac{ \pi }{2} \), what is the value of \( \Large tan \theta \)?

A). \( \Large \sqrt{7} \)

B). \( \Large \frac{7}{3} \)

C). 3

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

5). The value of \( \Large \frac{tan \ 27 ^{\circ} +cot \ 63 ^{\circ} }{tan \ 27 ^{\circ} (sin \ 25 ^{\circ} +cos \ 65 ^{\circ} )} \)

A). cosec 25 \(^{\circ}\)

B). 2 tan27\(^{\circ}\)

C). sin 25 \(^{\circ}\)

D). tan 65 \(^{\circ}\)

6). If \( \Large cos \theta \ge \frac{1}{2} \) in the first quadrant, which one of the following is correct?

A). \( \Large \theta \le \frac{ \pi }{3} \)

B). \( \Large \theta \ge \frac{ \pi }{3} \)

C). \( \Large \theta \le \frac{ \pi }{6} \)

D). \( \Large \theta \ge \frac{ \pi }{6} \)

7). If x=\( \Large a \ sec \theta \ cos \theta ,\ y=b \ sec \theta \ sin \theta \) and z=\( \Large c \ tan \theta \ \), then the value of \( \Large \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}} \) is

A). 9

B). 0

C). 1

D). 4

8). If \( \Large cos^{4} \theta -sin^{4} \theta =\frac{2}{3} \), then the value of \( \Large 1-2 \ sin^{2} \theta \) is

A). 0

B). \( \Large \frac{2}{3} \)

C). \( \Large \frac{1}{3} \)

D). \( \Large \frac{4}{3} \)

9). If \( \Large 7 sin^{2} \theta +3cos^{2} \theta =4 \) , then the value of \( \Large cos \theta (0 ^{\circ} \le \theta \le 90 ^{\circ} ) \) is

A). \( \Large \frac{\sqrt{6}}{2} \)

B). \( \Large \frac{\sqrt{3}}{2} \)

C). \( \Large \frac{\sqrt{2}}{2} \)

D). \( \Large \frac{\sqrt{5}}{2} \)

10). If \( \Large tan \ 8 \ \theta =cot \ 2 \ \theta \) where \( \Large 0< 8 \ \theta <\frac{ \pi }{2} \), then what is the value of \( \Large tan \ 5 \theta \) ?

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

B). 1

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

D). 0