Evaluate exactly.
-$\sec \left( \cos ^ { - 1 } \left( \frac { 1 } { 2 } \right) \right)$
A) 2
B) -1
C) $\frac { - 1 } { 2 }$
D) $- 2$

Question

Quizplus · Accepted Answer

Let $y = \cos ^ { - 1 } \left( \frac { 1 } { 2 } \right)$, then we have $\cos y = \frac12$.

Using the Pythagorean identity $\sin ^ 2 y + \cos ^ 2 y = 1$, we get $\sin y = \sqrt { 1 - \cos ^ 2 y } = \frac { \sqrt { 3 } } { 2 }$.

Therefore,

$$\sec \left( \cos ^ { - 1 } \left( \frac { 1 } { 2 } \right) \right) = \sec y = \frac { 1 } { \cos y } = \frac { 1 } { \frac { 1 } { 2 } } = 2.$$

Hence, the answer is \textbf{A) 2}.

Evaluate Exactly sec⁡(cos⁡−1(12))\sec \left( \cos ^ { - 1 } \left( \frac { 1 } { 2 } \right) \right)sec(cos−1(21​))