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Problems Set III

Example

If , show that

Solution

Given

Hence proved.


 

Example

If , prove that

Solution
Given

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.


 

Example

If prove that

Solution

Given that

Therefore

(or) (1)

To prove

Taking the L.H.S.

=

= From (1) above

=

=1 [By identity]

Hence proved.


 

Example

Prove that sec6θ= tan6θ + 3tan2θsec2θ + 1.

Solution

To prove: sec6θ= tan6θ + 3tan2θsec2θ + 1.

Taking the L.H.S.

=

=(using the corollary of the identity )

=

Applying the algebraic identity

== R.H.S.

Hence the result.


 

Example

Prove that

Solution

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.


 

Example

Prove that .

Solution

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.

 

 

Example

If and show that

Solution

LHS = m2-n2

=

=[Identity used =

RHS=

=

=

=

=

=

=

=

=

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


 

Example

Prove that

Solution

L.H.S.

=

=

=

R.H.S=

=

=

=

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

Hence proved.


 

Example

Prove that

Solution

L.H.S

=

=

=

=R.H.S.

Hence proved.


 

Example
If
Solution

To prove
 

L.H.S.

=

=

=

=1=R.H.S.

Hence proved.





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