# Problems Set II

Example

Find the numerical value of 2(sin^{6}Î¸ +cos^{6}Î¸) â€“ 3(sin^{4}Î¸ + cos^{4}Î¸) + 1

Solution

The algebraic identity

Writing sin^{2Î¸ } + cos^{2Î¸ } = 1, we have

Â

Example

Prove that

Solution

L.H.S.=

Taking conjugate of the denominator we have

= We know that ,

=

=

=

Therefore it is proved that ** **

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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

Â

=

=

= 2

Hence proved.

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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 (**) = **

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Example

Prove that** **

Solution

L.H.S. =** **

Writing tanÎ¸ in terms of sinÎ¸ and cosÎ¸ * ,*

Taking sin^{2Î¸ } as common factor,

= R.H.S.

Hence proved.