Question

Given sinx - cosx = $\frac{1}{2}$.

If 2sinxcosx = a, find the value of 16a.

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Answer 1

From the previous problem we found that, we can find 1+2sinxcosx if we square sinx+cosx. We will approach this problem in a similar way. Instead of sinx+cosx, we are given sinx-cosx.

Consider $(sinx-cosx{)}^2$ = $si{n}^{2}x$ +$co{s}^{2}x$ - 2sinxcosx

$\Rightarrow$ $\frac{1}{4}$= 1- 2sinxcosx

$\Rightarrow$ 2 sinxcosx = 1 - $\frac{1}{4}$

= $\frac{3}{4}$

$\Rightarrow$ a = $\frac{3}{4}$

$\Rightarrow$ 16a = 4 $\times$ 3

= 12

Key steps / concepts: (1) Guessing the operator to get sinx cosx from sinx-cosx (2) $si{n}^{2}x$ + $co{s}^{2}x$ = 1