Question

If 3 cosx+ 2cos3x = cosy, 3sinx+2sin3x=siny, then the value of $4(cos2x{)}^2$ is

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Answer 1

We want to find cos2x. There is no 2x in the two relations given. Instead, we have 3x and x. R.H.S of the two relations are cosy and siny. We don's want cosy or sinywhile finding cos2x.

We are familiar with eliminating cosy and siny when they are given separately. We square and add them. There is one more motive behind this. Those who have mentally calculated the steps, they will see that, cos3xcosx+sin3xsinx will be part of the expansion. This will give us cos2x.

$Co{s}^{2}y+si{n}^{2}y=9co{s}^{2}x+4co{s}^{2}3x+12cos3xcosx+9si{n}^{2}x+4si{n}^{2}3x+12sin3xsinx$

1 =9 + 4 + 12 (cos3x cosx + sin3x + sinx)

1= 13 + 12 (cos2x)

cos2x = -1

$4(cos2x{)}^2$ = 4

Key steps/concepts: (1) Eliminating siny and cosy

(2) Separating the terms which will lead to cos2x