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Directions : Read the following questions carefully and choose the right answer.
» Explain it
D
8 sin x – 4 = cos x ⇒ 8 sin x – 4 = 
1 – sin2 x

⇒ 1 – sin2 x = (8 sin x – 4)2

⇒ 1 – sin2 x = 64 sin2 x + 16 – 64 sin x

⇒ 65 sin2 x – 64 sin x + 15 = 0

⇒ 65 sin2 x – 39 sin x  –  25 sin x + 15 = 0

⇒ 13 sin x (5 sin x  – 3)  –  5 (5 sin x  –  3) = 0

⇒ (5 sin x – 3) (13 sin x – 5) =0

⇒ sin x =  3  or sin x =  5
5 13

Hence, option D is correct.

2
If cot ( π  –  Θ )
3
, then the value of sinΘ – cosΘ = ?
2 2
» Explain it
C
cot ( π  –  Θ )
3
2 2

cot ( π  –  Θ ) = cot 30°
2 2

or, 90° –  Θ  = 30°
2

or,  Θ  = 60°
2

∴ Θ = 120°

Now, sinΘ – cosΘ = sin120° – cos120°

= sin (90° + 30°) – cos (90°+30°)

= cos 30° + sin 30°

3
 +  1  = 
3 +1
2 2 2

Hence, option C is correct.

3
If cosΘ = –   1  and π < Θ  < 3π, find the value of 4 tan2Θ – 3 cosec2Θ
2
» Explain it
B
We know that

sinΘ  = ±  1 – cos2Θ

or,  sinΘ  = 
1–  1
4
 = – 
3
2

[ Since Θ lies in the third quadrant, value of sinΘ is negative.]

or, cosecΘ  =   –  2
3

and tanΘ  =  sinΘ  = 
3
cosΘ

[ Since Θ lies in the third quadrant, value of tanΘ is positive.]

Now, 4 tan2Θ – 3 cosec2Θ = 4 × 3 – 3 ×  4  = 8
3

Hence, option B is correct.

4
The value of  sin 300° tan 330° sec 420°  is :
cot 135° cos 210° cosec 315°
» Explain it
D
sin(360° – 60°) tan(360° – 30°) sec(360° + 60°)
cot(180° – 45°) cos(180° + 30°) cosec(360° – 45°)

( – sin 60° )( – tan 30°) sec 60°
(– cot 45°)(– cos 30°)(– cosec 45°)

= – (
3
 ×  1 × 2 )  = – 
2
3
2
3
1 ×
3
 × 2
2

Hence, option D is correct.

5
If tanΘ + cotΘ = 16, then find the ratio of tan2Θ + cot2Θ to tan2Θ + cot2Θ + 20 tanΘ.cotΘ
» Explain it
D
Given, tanΘ + cotΘ = 16

Squaring both sides, we get

tan2Θ + 2tanΘ cotΘ + cot2Θ = 256

or, tan2Θ + cot2Θ = 256 – 2

∴   tan2Θ + cot2Θ = 254

Now, tan2Θ + cot2Θ + 20tanΘ.cotΘ

= ( tan2Θ + cot2Θ )  +   20tanΘ 1
tanΘ

= 254 + 20 = 274

∴  Reqd. ratio  =  254  =  127  =   127 : 137
274 137

Hence, option D is correct.