How do you express $sin(pi/ 4 ) * sin( ( pi) / 3 )$ without using products of trigonometric functions?

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Answer 1 · 2016-05-20T05:54:08

$sin(pi/4)sin(pi/3)=1/2[cos(pi/12)+cos((5pi)/12)]$

We know that $cos(A+B)=cosAcosB-sinAsinB$ ..............(1)

and $cos(A-B)=cosAcosB+sinAsinB$ ..............(2)

Now subtracting (2) from (1)

$2sinAsinB=cos(A-B)-cos(A+B)$ or

$sinAsinB=1/2cos(A-B)-1/2cos(A+B)$

Hence, $sin(pi/4)sin(pi/3)=1/2cos((pi/4)-(pi/3))-1/2cos((pi/4)+(pi/3))$

= $1/2cos(-pi/12)-1/2cos((7pi)/12)$ ,

but $cos(-x)=cosx$ and $cos(pi-x)=-cosx$ ,

hence above is equal to

$1/2cos(pi/12)-1/2cos(pi-(5pi)/12)$

= $1/2cos(pi/12)+1/2cos((5pi)/12)=1/2[cos(pi/12)+cos((5pi)/12)]$

How do you express sin(pi/ 4 ) * sin( ( pi) / 3 ) sin(pi/ 4 ) * sin( ( pi) / 3 ) without using products of trigonometric functions?