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 four pi over five. Now the secant like cosine is an even function. So based on that identity here, I know that the secant of negative four pi over five is really just equal to the secant of positive four pi over five 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 expression. Now here you may have been tempted to to select the cosine of four pi over five as your correct answer, but if this had said one over the cosine of four pi over five, 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 four pi over five 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 )

12. Trigonometric Functions

Trigonometric Identities

Business Calculus

12. Trigonometric Functions

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

Step 1: Recall the definition of the secant function. The secant function is defined as the reciprocal of the cosine function: sec(x) = 1/cos(x). This means that sec(−4π/5) = 1/cos(−4π/5).

Step 2: Use the even property of the cosine function. The cosine function is even, meaning cos(−x) = cos(x). Therefore, cos(−4π/5) = cos(4π/5).

Step 3: Substitute the result from Step 2 into the secant function. Since sec(−4π/5) = 1/cos(−4π/5) and cos(−4π/5) = cos(4π/5), it follows that sec(−4π/5) = 1/cos(4π/5).

Step 4: Compare the given options. The correct answer must match the expression sec(−4π/5) = sec(4π/5), as derived in the previous steps.

Step 5: Verify the signs and relationships of the other options. For example, −cos(4π/5) and −sec(4π/5) involve negative signs, which do not match the derived equivalence. This confirms the correct answer is sec(4π/5).

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