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Sagot :
Step-by-step explanation:
To solve the given problem, we start with the equation \( 29 \cos \theta = 21 \).
First, solve for \( \cos \theta \):
\[
\cos \theta = \frac{21}{29}
\]
Next, find \( \sin \theta \) using the Pythagorean identity:
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
Substitute \( \cos \theta = \frac{21}{29} \):
\[
\sin^2 \theta + \left(\frac{21}{29}\right)^2 = 1
\]
\[
\sin^2 \theta + \frac{441}{841} = 1
\]
\[
\sin^2 \theta = 1 - \frac{441}{841}
\]
\[
\sin^2 \theta = \frac{841 - 441}{841}
\]
\[
\sin^2 \theta = \frac{400}{841}
\]
\[
\sin \theta = \sqrt{\frac{400}{841}} = \frac{20}{29}
\]
Now, calculate \( \tan \theta \):
\[
\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{20}{29}}{\frac{21}{29}} = \frac{20}{21}
\]
Next, find \( \sin \theta - \tan \theta \):
\[
\sin \theta - \tan \theta = \frac{20}{29} - \frac{20}{21}
\]
To subtract these fractions, find a common denominator:
\[
\sin \theta - \tan \theta = \frac{20 \cdot 21 - 20 \cdot 29}{29 \cdot 21} = \frac{420 - 580}{609} = \frac{-160}{609}
\]
Finally, find \( \frac{1}{\sin \theta - \tan \theta} \):
\[
\frac{1}{\sin \theta - \tan \theta} = \frac{1}{\frac{-160}{609}} = \frac{609}{-160} = -\frac{609}{160}
\]
Therefore, the value of \( \frac{1}{\sin \theta - \tan \theta} \) is \( \boxed{-\frac{609}{160}} \).
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