Answer :

(i)
To Prove: 

Proof:
 
  



                 =   

                  = 

               
Since,   

                 

                 

                   =

                    = RHS
Hence, proved


(ii)
To Prove: 
     

Proof:
      LHS = +

Use the identity sin2θ + cos2θ = 1

=

=

=

=

= 2 sec A

= RHS

(iii)

[Hint: Write the expression in terms of sin θ and Cos θ]

=





Use the formula a3 - b3 = (a2 + b2 + ab)(a - b)

Cancelling (sin θ - cos  θ) from numerator and denominator


As cosθ = 1/secθ and sinθ = 1/cosecθ

= 1 + sec θ cosec θ

= RHS

(iv)

[Hint: Simplify LHS and RHS separately]

LHS 

use the formula secA = 1/cosA

= cos A + 1

RHS

Use the identity sin2θ + cos2θ = 1

Use the formula a2 - b2 = (a - b) ( a + b )

= cos A + 1

LHS = RHS

(v) using the identity 

LHS

Dividing Numerator and Denominator by sin A


Use the formula cotθ = cosθ / sinθ

Using the identity 


Use the formula a2 - b2 = (a - b) ( a + b )



= cot A + cosec A

= RHS

(vi)

Dividing numerator and denominator of LHS by cos A

As cosθ = 1/secθ and tanθ = sinθ /cosθ

Rationalize the square root to get,

Use the formula a2 - b2 = (a - b) ( a + b ) to get,


Use the identity sec2θ = 1 + tan2θ to get,

= sec A + tan A

= RHS


(vii)
To Prove:


Proof:
LHS


Since 



As tanθ = sinθ/cosθ
= tanθ
= R.H.S
Hence, Proved.
(viii)
To Prove:

Proof:
LHS = (sin A + cosec A)2 + (cos A + sec A)2

Use the formula (a+b)2 = a2 + b2 + 2ab to get,

= (sin2A + cosec2A + 2sin A cosec A) + (cos2A + sec2A + 2 cos A sec A)
Since sinθ=1/cosecθ  and cosθ = 1/secθ

= sin2A + cosec2A + 2 + cos2A + sec2A + 2
=(sin2A + cos2A)+cosec2A+sec2A+2+2
Use the identities sin2A + cos2A = 1, sec2A = 1 + tan2A and cosec2A = 1 + cot2A  to get

= 1+ 1 + tan2A + 1 + cot2A + 2 + 2

= 1 + 2 + 2 + 2 + tan2A + cot2A

= 7 + tan2A + cot2A

= RHS

(ix)
To Prove:

[Hint: Simplify LHS and RHS separately]


Proof:
LHS = (cosec A – sin A) (sec A – cos A)
Use the formula sinθ = 1/cosecθ and cosθ = 1/secθ

     

     
     
      

       = cos A sin A

RHS
 
use the formula tanθ=sinθ/cosθ and cotθ=cosθ/sinθ



Use the identity sin2A + cos2A = 1

= cos A sin A

LHS = RHS

(x)
To Prove:
  

Proof:
Taking left most term
Since, cot A is the reciprocal of tan A, we have

= right most part
Taking middle part:

= right most part
 
Hence, Proved.

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