1 . You are asked to investigate the formulae of the sums and the products of trigonometric ratios .
Let n
� r� Fn ( ) � � sin sin , where n is a positive integer greater than 1 .
2n n r�1
( a ) Express F ( 2 ) as a single sine ratio .
( b ) Show that
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( i ) cos ( x � y) �cos ( x � y) � 2sin xsin y ;
B �A B �A ( ii ) cos A�cos B�
2sin sin 2 2
( c ) Hence , express F ( 4 ) as a single sine ratio .
( d ) Show that
Let
( 1 � n)
� Fn ( ) � sin .
2n
2� 2� z �cos
� isin , z � . n n
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Let n r
G( n) �� z �1 , where n is a positive integer . r�1 r
( e ) Is it true that z �1 � 2? Explain your answer .
( f ) Using ( d ), express Gn ( ) in a single cotangent ratio .
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