Given sec \( \theta=2 \), use trigonometric identities to find the exact value of the following expressions. \( \begin{array}{llll}\text { (a) } \cos \theta & \text { (b) } \tan ^{2} \theta & \text { (c) } \csc \left(90^{\circ}-\theta\right) & \text { (d) } \csc ^{2} \theta\end{array} \) \( \begin{array}{l}\text { (a) } \cos \theta=\square \\ \text { (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Type an } \\ \text { exact answer, using radicals as needed.) }\end{array} \)

To find the values of the expressions given that \( \sec \theta = 2 \), we can start by recalling that \( \sec \theta = \frac{1}{\cos \theta} \). Since secant is the reciprocal of cosine, we can find \( \cos \theta \) directly: (a) Since \( \sec \theta = 2 \), we have: \[ \cos \theta = \frac{1}{\sec \theta} = \frac{1}{2} \] Now, using the Pythagorean identity for tangent: (b) The identity \( \tan^2 \theta + 1 = \sec^2 \theta \) gives us: \[ \tan^2 \theta = \sec^2 \theta - 1 = 2^2 - 1 = 4 - 1 = 3 \] For the co-function identity involving cosecant: (c) The expression \( \csc(90^\circ - \theta) \) translates to \( \sec \theta \) due to the co-function identities: \[ \csc(90^\circ - \theta) = \sec \theta = 2 \] Finally, we can find \( \csc^2 \theta \) using the relationship between sine and cosecant: (d) Recall that \( \csc^2 \theta = 1 + \cot^2 \theta \) and since we know \( \cot^2 \theta = \frac{1}{\tan^2 \theta} \): \[ \csc^2 \theta = 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3} \] So the answers are: - \( \cos \theta = \frac{1}{2} \) - \( \tan^2 \theta = 3 \) - \( \csc(90^\circ - \theta) = 2 \) - \( \csc^2 \theta = \frac{4}{3} \)