# Problems Set I

Prove that

Proof:

L.H.S. =

Denominator = 1 â€“ cos Î¸

Its conjugate = 1 + cos Î¸

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

we have,

=

In the denominator(1-cosÎ¸)(1+cosÎ¸) = 1 - cos^{2}Î¸ using the algebraic identity

we know that ,1 â€“ cos^{2Î¸ } = sin^{2Î¸ }.

=

Hence proved.

ProveÂ Â the following identity:

Proof:

L.H.S. = Â Â Â Â Â Â

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

=

= [Using the identity 1 â€“ cos^{2Î¸ } = sin^{2Î¸ }]

=

=

=

= cosecÎ¸ - cotÎ¸

R.H.S.

Hence proved.

Â

Prove the following identity

Â

Proof:

L.H.S.

=

Changing all the as in this expression, we have

Applying the algebraic identity for , we get

Â

Â

Prove that

Â

Taking the L.H.S. we have,

=

(Replacing the 1 of the numerator by (sec^{2Î¸ } - tan^{2Î¸ }) using** **identity)

Canceling (1 - secÎ¸ + tanÎ¸ ) in the numerator and denominator as a single entity we have,

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

Hence proved.

Â

Prove that

Proof:

Taking the L.H.S.

Replacing we get,

=

=

Hence proved.

Â