2 . You are asked to investigate the integral of the product of trigonometric functions .
Let A and B be two unequal positive integers .
( a ) Show that cos ( A� B) x � cos ( A� B) x � 2cos Ax cos Bx .
( b ) Hence , evaluate � cos Ax cos Bxdx .
0
Let z �cos�� isin�
, z � .
�
[ 2 ]
[ 4 ]
( c ) ( i ) By considering the product of z and its conjugate , express 1 z in modulus-argument form .
( ii ) Hence , express cos� in terms of z .
( d ) By considering
3 1
� cos3 � isin 3 , show that
3 z � � cos � � ( cos3� � 3cos �)
.
4
( e ) For any positive odd number n , express cos n � in terms of n and � . [ 5 ]
( f ) Hence , evaluate
�
5 cos6 x cos d
0
� x x .
[ 5 ]
[ 4 ]
[ 5 ]
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