Direction: Study the following questions carefully and choose the right answer:
1
If sin A = 3/5 and A is an acute angle, then tan A + sec A is equal to
» Explain it
C
sin A =  3  
5

we know that,

cos2 A = 1 – sin2 A

∴ cos A = 
1 –  9
25
 = 
16
25
 =  4
5

∴   tan A + sec A =  sin A  +  1
cos A cos A

=  
sin A + 1
cos A
  =  
3  + 1
5
4
5

=   8  ×  5  = 2.
5 4

Hence, option C is correct.

2
What is the value of   tan A – sin A  ?
sin3 A
» Explain it
C
    sin A  – sin A
tan A – sin A   =   cos A
sin3 A sin3 A

  sin A  ( 1  – 1 )
=    cos A 
sin3 A

   (  1 – cos A  )
  =   cos A
sin2 A

=   ( 1 – cos A )
 cos A.sin2 A 

  =   1 – cos A
cos A (1 – cos2 A)

=   1 – cos A
cos A (1 – cosA) (1 + cos A)

=   1  .  1   =   sec A
cos A 1 + cos A 1 + cos A

Hence, option C is correct.

3
(1 – tan A)2 + (1 + tan A)2 + (1 – cot A)2 + (1 + cot A)2 is equal to
» Explain it
C
(1 – tan A)2 + (1 + tan A)2 + (1 – cot A)2 + (1 + cot A)2

= 1 + tan2 A – 2 tan A + 1 + tan2 A + 2 tan A + 1 + cot2 A – 2 cot A + 1 + cot2 A + 2 cot A

= 1 + tan2 A + 1 + tan2 A + 1 + cot2 A + 1 + cot2 A

= sec2 A + sec2 A + cosec2 A + cosec2 A

[     1 + tan2 A = sec2 A    and    1 + cot2 A = cosec2 A   ]

= 2 (sec2 A + cosec2 A)

=   [ 1  +  1 ]   =  2 [  sin2A + cos2 ]
 cos2 A   sin2 A  sin2 A.cos2 A 

=  2 ×  1
  =  2 sec2 A.cosec2A
 
sin2 A.cos2 A

Hence, option C is correct.

4
If α, β and γ are acute angles such that sin α = (3 / 2), cos β = (3 / 2) and tan γ = 1, then what is α + β + γ equal to?
» Explain it
C
sin α = 
3
 
2

⇒ sin α = sin 60°

⇒  α = 60°

cos β = 
3
 
2

⇒ cos β = cos 30°

 ⇒ β = 30°

and tan γ = 1   ⇒  γ = 45°

∴    α +β + γ = 60° + 30° + 45° = 135°.

Hence, option C is correct.

5
If cos A + cosA = 1, then what is the value of 2(sin2 A + sin4 A)?
» Explain it
B
cos A + cos2 A = 1

cos A = 1 – cos2 A = sin2 A         ...(i)

Now,  2(sin2 A + sin4 A) = 2(sin2 A + cos2 A)

[ from eq.(i) sin2 A = cos A, then sin4 A = cos2 A ]

= 2 × 1 = 2.

Hence, option B is correct.