## Trigonometry Questions for SSC/Railway Exams (25 – 02 – 2018)

Trigonometry Questions for SSC/Railway Exams (25 – 02 – 2018)
1. What will be the simplified value of (secA - cosA)² + (cosecA - sinA)² - (cotA - tanA)² ?
a) 0
b) 1
c) 1.2
d) 2
Solution: (secA - cosA)² + (cosecA - sinA)² - (cotA - tanA)²
= (sec²A + cos²A - 2 secA cosA) + (cosec²A + sin²A - 2 cosecA sinA) - (cot²A + tan²A - 2 cotA tanA)
= (sec²A + cos²A - 2) + (cosec²A + sin²A - 2) - (cot²A + tan²A - 2)
= sec²A - tan²A + cos²A + sin²A + cosec²A - cot²A - 2
= 3 – 2 = 1
2. What is the value of (tan ⁴ 60° - sin ⁴ 90°) - 2(tan²45° - 3cos0°)²?
a) 3
b) 2
c) 1
d) 0
Solution :
(tan ⁴ 60° - sin ⁴ 90°) - 2(tan²45° - 3cos0°)²
= ( √ 3 ⁴ - 1 ⁴ ) - 2(1² - 3*1)² = (9 - 1) - 2(1 - 3)² = 8 - 2*4 = 0
3. If sec θ + tan θ = 3, what is the value of tan θ ?
a) 4/3
b) 3/2
c) 5/2
d) 5/4
Solution :
We know that sec² θ - tan² θ = 1.
Applying the formula (a² - b²) = (a + b)(a - b), we get
(sec θ + tan θ )(sec θ - tan θ ) = 1
Given, sec θ + tan θ = 3   ...(i)
3 * (sec θ - tan θ ) = 1
(sec θ - tan θ ) = 1/3   ...(ii)
Solving equations (i) and (ii)
2tan θ = 3 - 1/3 2tan θ = 8/3 tan θ = 4/3
4. If A and B are acute angles, sin (A - B) = 1/2 and cos (A + B) = 1/2, what are are values of A and B respectively?
a) 30°, 30°
b) 45°, 15°
c) 30°, 15°
d) 60°, 30°
Solution :
sin (A - B) = 1/2
sin (A - B) = sin 30° A - B = 30°   (i)
cos (A + B) = 1/2 A + B = 60°   …(ii)
From (i) and (ii), A = 45°, B = 15°

5. What is the value of [cosec (90° - θ ) - sin (90° - θ )] [cosec θ - sin θ ] [tan θ + cot θ ]?
a) 0
b) 1
c) -1
d) 1/2
Solution :
[cosec (90° - θ ) - sin (90° - θ )] [cosec θ - sin θ ] [tan θ + cot θ ]
= (sec θ - cos θ ) (cosec θ - sin θ ) (tan θ + cot θ )
= (1/cos θ - cos θ ) (1/sin θ - sin θ ) (tan θ + 1/tan θ )
= {(1 - cos² θ )/cos θ } {(1 - sin² θ )/sin θ } {(1 + tan² θ )/tan θ }
= (sin² θ /cos θ ) (cos² θ /sin θ ) (sec² θ /tan θ ) = 1
6. If sin θ . cos θ = ½, what is the value of (sin θ - cos θ )?
a) 2
b) 1
c) 0
d) -1
Solution :
Given, sin θ . Cos θ = 1/2
2 . sin θ . cos θ = 1 sin 2 θ = sin 90° 2 θ = 90° θ = 45°
(sin θ - cos θ ) = sin 45° - cos 45° = 0
7. What is the value of (cot ⁴ θ - cosec ⁴ θ + cot² θ + cosec² θ )?
a) -1
b) 0
c) 1
d) 2
Solution :
(cot ⁴ θ - cosec ⁴ θ + cot² θ + cosec² θ )
= (cot² θ - cosec² θ ) (cot² θ + cosec² θ ) + cot² θ + cosec² θ
= -1 * (cot² θ + cosec² θ ) + cot² θ + cosec² θ
= - cot² θ - cosec² θ + cot² θ + cosec² θ = 0
8. If cos θ = 3/5, what is the value of sin θ . sec θ . tan θ ?
a) 3/4
b) 4/3
c) 9/16
d) 16/9
Solution :
Given, cos θ = 3/5
sin θ = 4/5 sin θ /cos θ = 4/3
sin θ . sec θ . tan θ = sin θ . (1/cos θ ) . (sin θ /cos θ ) = (sin θ /cos θ )² = (4/3)² = 16/9

9. If a . sin θ + b . cos θ = c, what is the value of a . cos θ - b . sin θ ?
a) ± (-a² + b² + c²)
b) ± (a² - b² - c²)
c) ± (a² + b² - c²)
d) ± (a² - b² + c²)
Solution :
Given, a . sin θ + b . cos θ = c
Let a . cos θ - b . sin θ = k
Squaring and adding both equations we get:
a² + b² = c² + k²
k = ± (a² + b² - c²)
10. If P . cos θ + Q . sin θ = 8 and P . sin θ - Q . cos θ = 5, what is the value of (P² + Q²)?
a) 39
b) 40
c) 13
d) 2√10
Solution :
Given, P . cos θ + Q . sin θ = 8
P² . cos² θ + Q² . sin² θ + 2PQ . sin θ . cos θ = 64   (i)
Also given, P . sin θ - Q . cos θ = 5
P² . sin² θ + Q² . cos² θ - 2PQ . sin θ . cos θ = 25   … (ii)