D. $$ 7 \frac{1}{3} \mathrm{~cm} $$ 37. If $$ \sin x=\frac{5}{13} $$ and $$ 0^{\circ} \leq x \leq 90^{\circ} $$ find the value of $$ (\cos x- an x) $$. $$ \begin{array}{lll} ext { A. } \frac{7}{13} & ext { B. } \frac{12}{13} & ext { C) } \frac{79}{156} \ ext { D. } \frac{209}{156}\end{array} $$

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D. $$ 7 \frac{1}{3} \mathrm{~cm} $$ 37. If $$ \sin x=\frac{5}{13} $$ and $$ 0^{\circ} \leq x \leq 90^{\circ} $$ find the value of $$ (\cos x-	an x) $$. $$ \begin{array}{lll}	ext { A. } \frac{7}{13} & 	ext { B. } \frac{12}{13} & 	ext { C) } \frac{79}{156} \ 	ext { D. } \frac{209}{156}\end{array} $$

UpStudy AI · Accepted Answer

Let 
$$
\sin x = \frac{5}{13}.
$$

Since 
$$
0^\circ \leq x \leq 90^\circ,
$$
the value of $$\cos x$$ is positive. We find $$\cos x$$ using the Pythagorean identity:

$$
\cos x = \sqrt{1 - \sin^2 x} = \sqrt{1 - \left(\frac{5}{13}\right)^2} = \sqrt{1 - \frac{25}{169}} = \sqrt{\frac{144}{169}} = \frac{12}{13}.
$$

Next, we calculate $$\tan x$$:

$$
\tan x = \frac{\sin x}{\cos x} = \frac{\frac{5}{13}}{\frac{12}{13}} = \frac{5}{12}.
$$

Now, we need to find $$\cos x - \tan x$$:

$$
\cos x - \tan x = \frac{12}{13} - \frac{5}{12}.
$$

To subtract these fractions, find a common denominator, which is $$156$$:

$$
\frac{12}{13} = \frac{12 \times 12}{13 \times 12} = \frac{144}{156},
$$
$$
\frac{5}{12} = \frac{5 \times 13}{12 \times 13} = \frac{65}{156}.
$$

Subtracting:

$$
\cos x - \tan x = \frac{144}{156} - \frac{65}{156} = \frac{144 - 65}{156} = \frac{79}{156}.
$$

Thus, the value of $$\cos x - \tan x$$ is $$\frac{79}{156}$$.
The value of $$ \cos x - \tan x $$ is $$ \frac{79}{156} $$, which corresponds to option C.

D.
37. If and

find the value of .

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D. 37. If and find the value of .