Direction: Study the following questions carefully and choose the right answer:
1
What is cosec (75° + Θ) – sec (15° – Θ) – tan (55° + Θ) + cot (35° – Θ) equal to?
» Explain it
B
cosec (75° + Θ) – sec (15° – Θ) – tan (55° + Θ) + cot (35° – Θ)
 
= cosec (75° + Θ) – sec [90° – (75° – Θ)] – tan (55° + Θ) + tan [90 – (55° – Θ)]
 
= cosec (75° + Θ) – cosec (75° + Θ) – tan (55° + Θ) + tan (55° + Θ) = 0.

Hence, option B is correct.
 
2
If sin Θ.cos Θ = 1/2, then what is sin6Θ + cos6Θ equal to? 
» Explain it
D
sin6Θ + cos6Θ = (sin2Θ)3 + (cos2Θ)3

= (sin2Θ + cos2Θ)(sin4Θ + cos4Θ – sin2Θcos2Θ)

= 1 [(sin2Θ + cos2Θ)2 – 3sin2Θcos2Θ]

= 1 – 3 ×  1  =  1 .
4 4

Hence, option D is correct.

3
If 2 cot Θ = 3, so   2 cos Θ – sin Θ  =
2 cos Θ + sin Θ

» Explain it
C
2 cot Θ = 3  ⇒  cot Θ =  3
2

=   2 cos Θ – sin Θ
2 cosΘ + sin Θ

[ Dividing by sin Θ in numerator & denominator]

=  
2. cos Θ  – 1
sin Θ
  =  
2 ×  3  – 1
2
2. cos Θ  + 1
sin Θ
2 ×  3  + 1
2

[  ∵   cos Θ  = cot Θ =   3  ]
sin Θ  2 

=   2   =   1
4 2

Hence, option C is correct.

4
In circular measure, the value of the angle 11° 15' is
» Explain it
A
11° 15'

=  11° +  15°
60

=  11° +  1  =  45°
4 4

[∵  180° = πc]

∴     45°  =  π  ×  45  =  πc
4 180 4 16

Hence, option A is correct.

5
If 29 tanΘ=31, then  1+2sinΘ cosΘ  =
1–2 sinΘ cosΘ

» Explain it
B
29 tan Θ = 31  ⇒  tan Θ =  31
29

Expression =  1 + 2sinΘ.cosΘ
1 – 2sinΘ.cosΘ

=   sin2 Θ + cos2Θ + 2sinΘ.cosΘ
sin2Θ + cos2Θ – 2sinΘ.cosΘ

=   (sinΘ + cosΘ)2
(sinΘ – cosΘ)2

=    
sinΘ  +  cosΘ
cosΘ cosΘ
2
  =   tanΘ + 1 2
tanΘ – 1  
 
sinΘ  –  cosΘ
cosΘ cosΘ
 

=   
31  + 1
29
2   =  
31 + 29
29
2
31  – 1
29
 
31 – 29
29
 

=   60 2   = (30)2 = 900.
2  

Hence, option B is correction.