Question

Given $\sin x+\cos x$ = $\frac{5}{4}$. If $1+2\sin x\cos x=a$, find the value of $32a$.

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Answer 1

We want to find $1+2\sin x\cos x$
where $\sin x+\cos x$ is given.
By doing some operations we should get $1+2\sin x\cos x$.
We get $2\sin x\cos x$ as one term in the expansion of $(\sin x+\cos x{)}^2$. This is one of the motivations to consider $(\sin x+\cos x{)}^2$.
Also ${\sin }^{2}x+{\cos }^{2}x=1$
Either way, we have to consider
$(\sin x+\cos x{)}^2$.
Then $(\sin x+\cos x{)}^2=(\frac{5}{4}{)}^2=\frac{25}{16}.$
$\Rightarrow {\sin }^{2}x+{\cos }^{2}x+2\sin x\cos x=\frac{25}{16}$
$\Rightarrow 1+2\sin x\cos x=\frac{25}{16}=a$
$\Rightarrow 32a=32\times \frac{25}{16}$ = 50.