# Problems Set III

If** , **show that ** **

Given ** **

Hence proved.

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If , prove that** **

To prove: ** **

Taking the L.H.S. ** **

Substituting the given data ** **in the above equation, we get

L.H.S. =

=

=

=[By identity]

=R.H.S.

Hence proved.

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If** **prove that** **

Given that ** **

Therefore ** **

(or) ** **(1)

To prove ** **

Taking the L.H.S.

=

= From (1) above

=

=1 [By identity]

Hence proved.

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Prove that sec^{6}Î¸= tan^{6}Î¸ + 3tan^{2}Î¸sec^{2}Î¸ + 1.

To prove: sec^{6}Î¸= tan^{6}Î¸ + 3tan^{2}Î¸sec^{2}Î¸ + 1.

Taking the L.H.S.

=

=(using the corollary of the identity )

=

Applying the algebraic identity

== R.H.S.

Hence the result.

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Prove that

To prove

L.H.S.

Taking conjugate of the denominator

Taking as common factor in the numerator and as common factor in the denominator, we have

=

=R.H.S.

Hence proved.

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Prove that** .**

To prove ** **

L.H.S **= **

For multiplication convenience rewriting as ** **

=

Changing the required term into sine and cosine we have

Writing we get

=R.H.S.

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If and show that

LHS = m^{2}-n^{2}

=

=[Identity used =

RHS=

=

=

=

=

=

=

=

=

Thus we have L.H.S. = R.H.S. i.e.

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Prove that** **

L.H.S. ** **

=

=

=

R.H.S=

=

=

=

Thus L.H.S=R.H.S

Hence proved.

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Prove that** **

L.H.S ** **

=

=

=

=R.H.S.

Hence proved.

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To prove ** **

Â

L.H.S. ** **

=

=

=

=1=R.H.S.

Hence proved.