tan(sec-1sqrt(2))
sec^-1 sqrt 2, the angle with a secant of sqrt 2, is the angle with cosine of 1/sqrt2, which is 45 degrees. The tangent of that angle is exactly 1.
Another angle with a sec of sqrt2 is 315 degrees. The tangent of that angle is -1. So there are two possible answers.
Thank you so much, drwls. I understand it now!
To evaluate the expression tan(sec^(-1)(√2)), we can use trigonometric identities and inverse trigonometric functions.
1. First, let's rewrite the expression using the identity tan(x) = sin(x) / cos(x):
tan(sec^(-1)(√2)) = sin(sec^(-1)(√2)) / cos(sec^(-1)(√2))
2. Next, let's work with the inner function sec^(-1)(√2). This means we need to find the angle whose secant is √2. Since sec(x) = 1/cos(x), we can find the angle by taking the inverse cosine of 1/√2:
sec^(-1)(√2) = cos^(-1)(1/√2)
3. The angle whose cosine is 1/√2 can be determined using a reference right triangle. In a right triangle, the adjacent side divided by the hypotenuse gives the cosine of the angle. In this case, the adjacent side is 1, and the hypotenuse is √2. Therefore, the angle is π/4 radians or 45 degrees.
4. Now, let's substitute the angle into the trigonometric expression:
tan(sec^(-1)(√2)) = sin(π/4) / cos(π/4)
5. Since sin(π/4) = cos(π/4) = 1/√2, we can simplify the expression:
tan(sec^(-1)(√2)) = (1/√2) / (1/√2)
6. Dividing the numerators and denominators gives:
tan(sec^(-1)(√2)) = 1
Therefore, tan(sec^(-1)(√2)) equals 1.