Trigonometry Questions for SSC CHSL Exam PDF

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Trigonometry Questions for SSC CHSL Exam PDF
Trigonometry Questions for SSC CHSL Exam PDF

Trigonometry Questions for SSC CHSL Exam PDF:

SSC CHSL Trignometry Questions download PDF based on previous year question paper of SSC CHSL exam. 25 Very important Trignometry questions for SSC CHSL Exam.

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Question 1: If $\sin x+\frac{1}{\sin x}=2$ then $\cos ^5 x+\cot ^{5} x=?$

a) 0
b) -1
c) 1
d) 2

Question 2: If $cos^2x + (1 + sinx)^2 = 3$, then find the value of cosec x + sin(60+x), where 0º < x < 90º?

a) 3
b) 2
c) $2\frac{1}{2}$
d) $3\frac{1}{2}$

Question 3: If $x$ and $y$ are acute angles such that their sum is less than $90^{\circ}$ . If $\cos(2x-20^{\circ})=\sin(2y+20^{\circ})$ , then the value of $\tan(x+y)$ is

a) $0$
b) $1$
c) $\frac{1}{\sqrt{3}}$
d) $\sqrt{3}$

Question 4: If x is an acute angle such that
Tan (4x – 50°) = Cot ( 50° – x), then the value of x would be?

a) 60
b) 45
c) 50
d) 30

Question 5: If cosec 9x= sec 9x (0 <x<10), what is the value of x?

a) 9°
b) 3°
c) 5°
d) 6°

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Question 6: If $\sin x+1=\frac{2}{\sin x} $ then $\cos ^3 x+ \mathrm{cosec} ^{-3} x=?$

a) 0
b) -1
c) 1
d) 2

Question 7: If $\frac{sin2x}{sinx} + cos2x + sin^2x = 0$, find the value of x, if $0 \leq x \leq 180$.

a) 45º
b) 90º
c) 180º
d) More than one value is possible

Question 8: If $\frac{sin^2 \Theta}{4} = \frac{cos^2 \Theta}{9}$. Find the value of $tan^2 \Theta – cot^2 \Theta$.

a) 1/6
b) 13/27
c) 1/18
d) None of these

Question 9: If $\sec^4 x-\tan^4 x=4+4sin^2 x$ then $ x=?$ (0<x<90°)

a) 45°
b) 30°
c) 75°
d) 60°

Question 10: Simplify the expression $\sqrt{\frac{1+cosx}{1-cosx}}$.

a) sec x + tan x
b) sin x
c) cosec x + cot x
d) cos x

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Question 11: If $sinx = 1 – cos x$, find the value of cos 2x.

a) 0
b) 1
c) $\frac{1}{2}$
d) $\frac{\sqrt{3}}{2}$

Question 12: If it is known that
$\tan\theta + \cot\theta = 2$, then find the value of $\tan^3\theta + \cot^3\theta$

a) $\sqrt{5}$
b) 2
c) $1 + \sqrt{3}$
d) $2 + \sqrt{3}$

Question 13: If $\sin x+\frac{1}{\sin x}=\frac{5}{2}$ then $\tan {x}+\cot {x}=?$

a) 0
b) $\frac{2}{\sqrt3}$
c) $\frac{4}{\sqrt3}$
d) $\frac{\sqrt3}{2}$

Question 14: If $\cos x+\frac{1}{\cos x}=2$ then $\sin ^{5} x+\sec ^{-5} x=?$

a) -1
b) 0
c) 1
d) 2

Question 15: If $\tan x+\frac{1}{\tan x}=2$ then $\sin ^{5} x+\cos ^{5} x=?$

a) $\frac{1}{4\sqrt{2}}$
b) $\frac{1}{2\sqrt{2}}$
c) $\frac{1}{\sqrt{2}}$
d) $\frac{1}{8\sqrt{2}}$

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Question 16: The maximum value of sin θ + cos θ is

a) $1$
b) $\sqrt{2}$
c) $2$
d) $3$

Question 17: If $sin \Theta + cosec \Theta = 5$, Find the value of $sin^3 \Theta + cosec ^3 \Theta $.

a) 96
b) 115
c) 110
d) None of these

Question 18: The value of $cos^2 30^{\circ} + sin^2 60^{\circ} + tan^2 45^{\circ} + sec^2 60^{\circ} + cos0^{\circ}$ is

a) $4\frac{1}{2}$
b) $5\frac{1}{2}$
c) $6\frac{1}{2}$
d) $7\frac{1}{2}$

Question 19: If $cos x + cos^{2} x = 1,$ then $sin^{8} x + 2 sin^{6} x + sin^{4}$ x is equal to

a) 0
b) 3
c) 2
d) 1

Question 20: If $x= p$ $cosec θ$ and $y= q$ $cot θ$, then the value of $\frac{x^2}{p^2}-\frac{y^2}{q^2}$ is

a) $sin^{2} θ$
b) $tanθ$
c) $1$
d) $0$

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Question 21: Find the value of tan 4° tan 43° tan 47 tan 86°

a) $\frac{2}{3}$
b) $1$
c) $\frac{1}{2}$
d) $2$

Question 22: If $x cosθ – sinθ = 1,$ then $x^{2} – (1 +x^{2}) sinθ$ equals

a) 2
b) 1
c) – 1
d) 0

Question 23: If $sin θ + sin^{2} θ = 1$ then $cos^2 θ + cos^4 θ$ is equal to

a) None
b) 1
c) $Sin θ /cos^{2} θ$
d) $cos^{2}θ / Sin θ$

Question 24: The value of tan1°tan2°tan3° ……………tan89° is

a) 1
b) -1
c) 0
d) None of the options

Question 25: If 0° ≤ A ≤ 90°, the simplified form of the given expression sin A cos A (tan A – cot A) is

a) 1
b) 1 – 2 $sin^2$ A
c) 2 $sin^2$ A – 1
d) 1 – cos A

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Answers & Solutions:

1) Answer (A)

$\sin x+\frac{1}{\sin x}=2$
=>$\sin^2 x+1=2\sin x$
=>$(\sin x-1)^2=0$
=>$\sin x = 1$
=>$\cos x = 0$
The given expression would be,
$\cos ^5 x+\cot ^{5} x$
$\cos ^5 x+\frac{\cos^5 x}{\sin^5 x}$
$=0$

2) Answer (A)

$cos^2x + (1 + sinx)^2 = 3$
=> $cos^2x + 1 + sin^2x + 2sinx = 3$ (Since $ cos^2x + sin^2x = 1$)
=> $2 sin x = 1$ or sin x = ½ which means x=30 degrees [Solution in the first quadrant]
=> cosec x = 1/sin x = 2 and sin (60+x) = sin 90 = 1
=> The value of the expression is 2+1 = 3

3) Answer (B)

Given that , $x+y<90^{\circ} \dots (1)$
$\implies\cos(2x-20^{\circ})=\cos(90^{\circ}-(2y+20^{\circ}))$
Using the given property in $(1)$
$\implies 2x-20^{\circ}=70^{\circ}-2y$
$\implies 2x+2y=90^{\circ}$
$\implies x+y=45^{\circ}$
$\implies \tan(x+y)=1 $
Hence , the correct option is B

4) Answer (D)

We know that Tan ( 90 – x) = Cot x and Cot (90 – x) = Tanx
So 4x – 50 = 90 – y
Then 50 – x will be y
=> y = 50 – x
x + y = 50
Moreover,
90 – y = 4x – 50
=> 4x + y = 140
Subtracting the two equations, we get
3x = 90
=> x = 30
Hence the correct answer is option D.

5) Answer (C)

If cosec 9x= sec 9x,
Then $\sin 9x=\cos 9x$
In the first quadrant cos and sin are equal when $9x =45$°
Hence, x = 5°

6) Answer (C)

$\sin x+1=\frac{2}{\sin x} $
=>$\sin^2 x+\sin x=2$
=>$(\sin x-1)(\sin x+2)=0$
As $\sin x$ can only take values between -1 and 1,
=>$\sin x = 1$
=>$\cos x = 0$
The given expression would be,
$\cos ^3 x+ \mathrm{cosec}^{-3} x$
$\cos ^3 x + \sin^3 x$
$=1$

7) Answer (B)

$\frac{sin2x}{sinx} + cos2x + sin^2x = 0$
We know that $sin2x = 2 sinx cosx$ and $cos2x=cos^2x-sin^2x$
=> $2cosx + cos^2x = 0$
=> cos x =0 or cos x =2
Now, cos x =2 is not possible.
=> cos x = 0
In the range, $0 \leq x \leq 180$, cos x =0 at only 90º
Thus, B is the correct answer.

8) Answer (D)

$\frac{sin^2 \Theta}{4} = \frac{cos^2 \Theta}{9}$
==> $\frac{sin^2 \Theta}{cos^2 \Theta}$ = $\frac{4}{9}$
==> $tan^2 \Theta = \frac{4}{9}$
==> $cot^2 \Theta = \frac{9}{4}$
==> $tan^2 \Theta – cot^2 \Theta$ = 4/9 – 9/4 = -65/36
Since there is no such option, the correct option to choose is D.

9) Answer (D)

$\sec^4 x-\tan^4 x=4+4sin^2 x$
$(sec^2 x-\tan^2 x)( sec^2 x+\tan^2 x)=4+4sin^2 x$
$(1)( sec^2 x+\tan^2 x)=4+4sin^2 x$ (as $sec^2 x-\tan^2 x = 1$)
$\frac{(1+ sin^2 x)}{cos^2 x} = 4(1+sin^2 x) $
$cos^2 x = \frac{1}{4}$
$ cos x = \frac{1}{2} or \frac{-1}{2}$
Here, $cosx$ cannot be negative as (0<x<90°)

Hence, $ cos x = \frac{1}{2}$ and $ x = 60$°

10) Answer (C)

$\sqrt{\frac{1+cosx}{1-cosx}}$
Multiplying the numerator and denominator by 1+cosx, we get
$\sqrt{\frac{(1+cosx)^2}{(1-cosx)(1+cosx)}}$
=> $\sqrt{\frac{(1+cosx)^2}{(1- cos^2x}}$
=> $\frac{1+cosx}{sinx}$ = cosecx + cotx$

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11) Answer (B)

$sinx = 1 – cos x$
=> $sinx + cos x = 1 $
Squaring both sides we get,
$sin^2x + cos^2x + 2 sinx cos x = 1$
=> $1 + sin2x = 1$
=> sin 2x = 0 or x=0
=> cos 2x = 1

12) Answer (B)

$\tan\theta + \cot\theta = 2$
Cubing both sides we get
$\tan^3\theta + \cot^3\theta + 3\tan\theta*\cot\theta(\tan\theta+\cot\theta) = 8$
=> $\tan^3\theta + \cot^3\theta + 3*2 = 8$
=> $\tan^3\theta + \cot^3\theta = 2$

13) Answer (C)

$\sin x+\frac{1}{\sin x}=\frac{5}{2}$
=>$2\sin^2 x-5\sin x+2=0$
=>$(\sin x-1/2)(\sin x-2)=0$
=>$\sin x = 1/2$
=>$x=30$ degrees
=>$\tan x = \frac{1}{\sqrt3}$
=>$\cot x = \sqrt3$
The given expression would be,
$\tan {x}+\cot {x}$
$=\frac{1}{\sqrt3}+\sqrt3$
$=\frac{4}{\sqrt3}$

14) Answer (C)

$\cos x+\frac{1}{\cos x}=2$
=>$\cos^2 x+1=2\cos x$
=>$(\cos x-1)^2=0$
=>$\cos x = 1$
=>$\sin x = 0$
The given expression is,
$\sin ^5 x+\sec ^{-5} x$
$= \sin ^5 x+\frac{1}{\cos^5 x}$
$=0+1 = 1$

15) Answer (B)

$\tan x+\frac{1}{\tan x}=2$
=>$\tan^2 x+1=2\tan x$
=>$(\tan x-1)^2=0$
=>$\tan x = 1$
=>$\sin x = \frac{1}{\sqrt{2}}$
=>$\cos x = \frac{1}{\sqrt{2}}$
The given expression would be,
$ \sin ^5 x+\cos ^{5} x$
$ \sin ^5 x+\cos^5 {x}$
$=\frac{1}{4\sqrt{2}}+\frac{1}{4\sqrt{2}} = \frac{1}{2\sqrt{2}}$

16) Answer (B)

for asin θ + bcos θ + c,
maximum value = $c+\sqrt{a^{2}+b^{2}}$
minimum value = $c-\sqrt{a^{2}+b^{2}}$
for sin θ + cos θ , a = 1, b = 1, c = 0
maximum value = $c+\sqrt{a^{2}+b^{2}}=0+\sqrt{1^{2}+1^{2}}=\sqrt{2}$
so the answer is option B.

17) Answer (C)

$sin \Theta + cosec \Theta = 5$
==> $(sin \Theta + cosec \Theta)^3 = 5^3$
==>$ sin^3 \Theta + cosec ^3 \Theta + 3*sin \Theta * cosec \Theta (sin \Theta + cosec \Theta)$ = 125
==> $sin^3 \Theta + cosec ^3 \Theta + 3*5 = 125$
==> $sin^3 \Theta + cosec ^3 \Theta $ = 110.

18) Answer (D)

Substituting values of angles, we get,
3/4 + 3/4+ 1 + 4 + 1 = 7.5. Option D is the right answer.

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19) Answer (D)

$cos x + cos^2 x = 1$
=> $cos x = 1 – cos^2 x$
=> $cos x = sin^2 x$
$\therefore$ $sin^{8} x + 2 sin^{6} x + sin^{4} x$
= $(sin^4 x + sin^2 x)^2$
= $((cos x)^2 + sin^2 x)^2$
= $(cos^2 x + sin^2 x)^2 = 1$

20) Answer (C)

$x= p$ $cosec θ$
=> $cosec\theta$ = $\frac{x}{p}$
Also, $y= q$ $cot θ$
=> $cot\theta$ = $\frac{y}{q}$
$\because$ $cosec^2\theta-cot^2\theta$ = 1
=> $\frac{x^2}{p^2}$ – $\frac{y^2}{q^2}$ = 1

21) Answer (B)

Expression : tan 4° tan 43° tan 47 tan 86°
$\because$ $tan(90-\theta) = cot\theta$
=> $tan 4^{\circ} = tan(90^{\circ}-86^{\circ}) = cot 86^{\circ}$
Similarly, $tan 43^{\circ} = cot 47^{\circ}$
=> $(cot 86^{\circ} \times tan 86^{\circ}) * (tan 47^{\circ} \times cot 47^{\circ})$
Using, $tan\theta cot\theta$ = 1
=> 1*1 = 1

22) Answer (B)

Expression : $x cosθ – sinθ = 1$
=> $x = \frac{1}{cos\theta} + \frac{sin\theta}{cos\theta}$
=> $x = sec\theta + tan\theta$ ————–Eqn(1)
$\because$ $sec^2\theta – tan^2\theta = 1$
=> $(sec\theta + tan\theta)(sec\theta – tan\theta) = 1$
=> $(sec\theta – tan\theta) = \frac{1}{x}$ ————–Eqn(2)
Adding eqns(1)&(2)
=> $2sec\theta = x + \frac{1}{x} = \frac{x^2 + 1}{x}$
=> $sec\theta = \frac{x^2 + 1}{2x}$
Subtracting eqn(2) from (1)
=> $2tan\theta = x – \frac{1}{x} = \frac{x^2 – 1}{x}$
=> $tan\theta = \frac{x^2 – 1}{2x}$
We know that, $sin\theta = \frac{tan\theta}{sec\theta}$
=> $sin\theta = \frac{x^2 – 1}{2x} * \frac{2x}{x^2 + 1}
=> $sin\theta = \frac{x^2 – 1}{x^2 + 1}$
To find : $x^2 – (1 + x^2) sinθ $
= $x^2 – (1 + x^2) * \frac{x^2 – 1}{x^2 + 1}$
= $x^2 – (x^2 – 1)$
= 1

23) Answer (B)

Expression : $sin\theta + sin^2\theta = 1$
=> $sin\theta = 1 – sin^2\theta$
=> $sin\theta = cos^2\theta$
To find : $cos^2\theta + cos^4\theta$
= $cos^2\theta + (cos^2\theta)^2$
= $cos^2\theta + sin^2\theta$
= 1

24) Answer (A)

Expression : tan1°tan2°tan3° ……………tan88°tan89°
$\because$ $tan(90^{\circ}-\theta) = cot\theta$
=> tan 89° = tan(90°-1) = cot 1°
Similarly, tan 88° = cot 2° and so on till tan 46° = cot 44°
=> (tan1°tan2°tan3°…….tan45°……cot3°cot2°cot1°)
Using, $tan\theta cot\theta$ = 1 and $tan45^{\circ}$ = 1
=> 1*1 = 1

25) Answer (C)

Expression : $sin A cos A (tan A – cot A)$
= $sin A cos A (\frac{sin A}{cos A} – \frac{cos A}{sin A})$
= $sin A cos A (\frac{sin^2 A – cos^2 A}{sin A cos A})$
= $sin^2 A – cos^2 A$
= $sin^2 A – (1 – sin^2 A)$
= $2sin^2 A – 1$

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