If \(\frac{\sin θ-2 \sin ^{3} θ}{2 \cos ^{3} θ-\cos θ}=\frac{1}{\sqrt{7}}\), 0° < θ < 90°, then what is the value of (sec2 θ + cosecθ) ÷ (sec2 θ - cosecθ)?

This question was previously asked in
AAI Junior Assistant (Fire Service) Official Paper (Held On: 15 Nov, 2022 Shift 2)
View all AAI Junior Assistant Papers >
  1. \(1 \frac{2}{3}\)
  2. \(1 \frac{1}{8}\)
  3. \(-1 \frac{1}{4}\)
  4. \(-1 \frac{1}{3}\)

Answer (Detailed Solution Below)

Option 4 : \(-1 \frac{1}{3}\)
Free
ST 1: English
2.6 K Users
20 Questions 20 Marks 25 Mins

Detailed Solution

Download Solution PDF

Given:

\(\frac{\sin θ-2 \sin ^{3} θ}{2 \cos ^{3} θ-\cos θ}=\frac{1}{\sqrt{7}}\), 0° < θ < 90°

Formula used:

tan θ = sin θ / cos θ

sin2 θ + cos2 θ = 1

sec2 θ = 1 + tan2 θ

cosec2 θ = 1 + cot2 θ = 1 + 1/tan2 θ

Calculation:

\(\frac{\sin θ(1-2 \sin ^{2} θ)}{\cos θ(2 \cos ^{2} θ-1)}=\frac{1}{\sqrt{7}}\)

\(\frac{\sin θ(1-2(1-\cos^2 θ))}{\cos θ(2 \cos ^{2} θ-1)}=\frac{1}{\sqrt{7}}\)

\(\frac{\sin θ(2\cos^2 θ-1)}{\cos θ(2 \cos ^{2} θ-1)}=\frac{1}{\sqrt{7}}\)

\(\frac{\sin θ}{\cos θ}=\frac{1}{\sqrt{7}}\)

tan θ = 1/√7

sec2 θ = 1 + tan2 θ = 1 + (1/7) = 8/7

cosec2 θ = 1 + cot2 θ = 1 + 7 = 8

Now, substitute into the expression:

(sec2 θ + cosec2 θ) / (sec2 θ - cosec2 θ)

= (8/7 + 8) / (8/7 - 8)

= (8 + 56) / (8 - 56)

= 64 / (-48)

= -64 / 48

= -4 / 3 = \(-1 \frac{1}{3}\)

∴ The value of (sec2 θ + cosec2 θ) / (sec2 θ - cosec2 θ) is \(-1 \frac{1}{3}\).

Latest AAI Junior Assistant Updates

Last updated on Apr 24, 2025

-> The AAI Junior Assistant Response Sheet 2025 has been out on the official portal for the written examination.

-> AAI has released 168 vacancies for Western Region. Candidates had applied online from 25th February to 24th March 2025.

-> A total number of 152 Vacancies have been announced for the post of Junior Assistant (Fire Service) for Northern Region.

-> Eligible candidates can apply from 4th February 2025 to 5th March 2025. 

-> Candidates who have completed 10th with Diploma or 12th Standard are eligible for this post.

-> The selection process includes a Computer Based Test, Document Verification, Medical Examination (Physical Measurement Test), Driving Test and a Physical Endurance Test.

-> Prepare for the exam with AAI Junior Assistant Previous year papers.

Get Free Access Now
Hot Links: teen patti gold teen patti master apk download teen patti rummy 51 bonus teen patti star login teen patti noble