Question

If $x$ is acute and $3\sin x=4\cos x$, Find the value of $\frac{(3+3\sin x)(1-\sin x)}{(1+\cos x)(1-\cos x)}$

Answer 1

Given that $x$ is acute angle and $3\sin x=4\cos x$

$3\sin x=4\cos x$

$\Rightarrow \frac{\sin x}{\cos x}=\frac{4}{3}$

$\Rightarrow \tan x=\frac{4}{3}$

$\Rightarrow \cot x=\frac{3}{4}$ ……. (1)

Consider given function,

$\frac{(3+3\sin x)(1-\sin x)}{(1+\cos x)(1-\cos x)}$

$=\frac{3(1+\sin x)(1-\sin x)}{(1+\cos x)(1-\cos x)}$

$=\frac{3(1-{\sin }^{2}x)}{(1-{\cos }^{2}x)}$

$=\frac{3{\cos }^{2}x}{{\sin }^{2}x}$

$=3{\cot }^{2}x$

By equation (1)

$=3\times {\left(\frac{3}{4}\right)}^2$

$=\frac{27}{16}$