Select the expression with the same value as the given expression.sec(−4π5)\sec\left(-\frac{4\pi}{5}\right)sec(−54π)

sec ⁡ ( 4 π 5 ) \sec\left(\frac{4\pi}{5}\right) sec ( 5 4 π )

Select the expression with the same value as the given expression. sec ( − 4 π 5 ) \sec\left(-\frac{4\pi}{5}\right) sec ( − 5 4 π )

Here we're asked to select the expression with the same value as the given expression and our given expression here is the secant of negative 4 pi over 5. Now the secant, like cosine, is an even function. So based on that identity here, I know that the secant of negative 4pi over 5 is really just equal to the secant of positive 4 pi over 5 because this function is even. So I see that that's one of my choices here. So I know that this one has the same value as my given pi over 5 as your correct answer. But if this had said 1 over the cosine of 4 pi over 5, that would also be correct, but it doesn't. Now our negative answers are clearly not correct because secant is an even function. So we are only left with our final answer as the secant of 4pi over 5 as our correct choice. Thanks for watching, and I'll see you in the next one.

Select the expression with the same value as the given expression. sec ⁡ ( − 4 π 5 ) \sec\left(-\frac{4\pi}{5}\right) sec ( − 5 4 π )

sec ⁡ ( 4 π 5 ) \sec\left(\frac{4\pi}{5}\right) sec ( 5 4 π )

cos ⁡ ( 4 π 5 ) \cos\left(\frac{4\pi}{5}\right) cos ( 5 4 π )

− cos ⁡ ( 4 π 5 ) -\cos\left(\frac{4\pi}{5}\right) − cos ( 5 4 π )

− sec ⁡ ( 4 π 5 ) -\sec\left(\frac{4\pi}{5}\right) − sec ( 5 4 π )

6. Trigonometric Identities and More Equations

Introduction to Trigonometric Identities

Trigonometry

6. Trigonometric Identities and More Equations

Introduction to Trigonometric Identities

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Multiple Choice

Select the expression with the same value as the given expression.
$\sec\left(-\frac{4\pi}{5}\right)$

$\cos\left(\frac{4\pi}{5}\right)$

$-\cos\left(\frac{4\pi}{5}\right)$

$\sec\left(\frac{4\pi}{5}\right)$

$-\sec\left(\frac{4\pi}{5}\right)$

Verified step by step guidance

Understand that the secant function, $ \sec(x) $, is the reciprocal of the cosine function, $ \cos(x) $. Therefore, $ \sec(x) = \frac{1}{\cos(x)} $.

Recognize that the secant function is an even function, meaning $ \sec(-x) = \sec(x) $. This property will help simplify $ \sec\left(-\frac{4\pi}{5}\right) $ to $ \sec\left(\frac{4\pi}{5}\right) $.

Evaluate the cosine function at $ \frac{4\pi}{5} $. Since $ \cos\left(\frac{4\pi}{5}\right) $ is the cosine of an angle in the second quadrant, it will be negative.

Use the reciprocal identity to find $ \sec\left(\frac{4\pi}{5}\right) $ from $ \cos\left(\frac{4\pi}{5}\right) $. Since $ \cos\left(\frac{4\pi}{5}\right) $ is negative, $ \sec\left(\frac{4\pi}{5}\right) $ will also be negative.

Compare the expressions: $ \sec\left(-\frac{4\pi}{5}\right) $ simplifies to $ \sec\left(\frac{4\pi}{5}\right) $, which matches the expression $ -\sec\left(\frac{4\pi}{5}\right) $ when considering the negative sign from the cosine function in the second quadrant.

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