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Problems Set II

Example

Find the numerical value of 2(sin6θ +cos6θ) – 3(sin4θ + cos4θ) + 1

Solution

The algebraic identity

Writing sin2θ + cos2θ = 1, we have


 

Example

Prove that

Solution

L.H.S.=

Taking conjugate of the denominator we have

= We know that ,

=

=

=

Therefore it is proved that


 

Example

Show that

Solution

Taking the L.H.S. and multiplying both the terms we get,

Changing all the terms in terms of sine and cosine, we get

 

=

=

= 2

Hence proved.


 

Example

If =p, evaluate the value of .

Solution

Given : =p,

We know by the corollary of the identity

Writing as ()(), we get

()(),=1

But from the given data =p. Substituting in the above equation, we get

p() = 1

Hence () =

 

 

Example

Prove that

Solution

L.H.S. =

Writing tanθ in terms of sinθ and cosθ ,

Taking sin2θ as common factor,

= R.H.S.

Hence proved.





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