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Problems Set I

Example

Prove that

Solution

Proof:

L.H.S. =

Denominator = 1 – cos θ

Its conjugate = 1 + cos θ

Multiplying and dividing L.H.S. by (1 + cosθ )

we have,

=

In the denominator(1-cosθ)(1+cosθ) = 1 - cos2θ using the algebraic identity

we know that ,1 – cos2θ = sin2θ .

=

Hence proved.

 
Example

Prove  the following identity:

Solution

Proof:

L.H.S. =       

Taking the conjugate of the denominator as explained earlier inside the square root,

=

= [Using the identity 1 – cos2θ = sin2θ ]

=

=

=

= cosecθ - cotθ

R.H.S.

Hence proved.


 

Example

Prove the following identity
 

Solution

Proof:

L.H.S.

=

Changing all the as in this expression, we have


Applying the algebraic identity for , we get

 

 

Example

Prove that
 

Solution

Taking the L.H.S. we have,

=

(Replacing the 1 of the numerator by (sec2θ - tan2θ ) using identity)

Canceling (1 - secθ + tanθ ) in the numerator and denominator as a single entity we have,

Taking conjugate of the numerator (1 + sinθ ) we get,

Hence proved.


 

Example

Prove that

Solution

Proof:

Taking the L.H.S.

Replacing we get,

=

=

Hence proved.

 





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