Question

Simplest form of \(\frac{\sec A}{\sqrt{\sec^2 A - 1}}\) is

• A. \(\sin A\)
• B. \(\tan A\)
• C. \(\csc A\)
• D. \(\cos A\)

Hint:

Whenever you see \(\sqrt{\sec^2 A - 1}\) or \(\sqrt{1 - \sin^2 A}\), immediately replace them using basic identities (\(\tan A\) and \(\cos A\) respectively) to simplify.

Answer 1

The Correct Option is C

To simplify the expression \(\frac{\sec A}{\sqrt{\sec^2 A - 1}}\), we start by using a fundamental trigonometric identity:

• \(\sec^2 A - 1 = \tan^2 A\)

Substituting this identity into the original expression, we have:

\(\frac{\sec A}{\sqrt{\sec^2 A - 1}} = \frac{\sec A}{\sqrt{\tan^2 A}} = \frac{\sec A}{\tan A}\)

Now, recall that:

• \(\tan A = \frac{\sin A}{\cos A}\)
• Thus, \(\frac{1}{\tan A} = \frac{\cos A}{\sin A}\)

Therefore, substituting back, we get:

\(\frac{\sec A}{\tan A} = \sec A \times \frac{\cos A}{\sin A}\)

Since \(\sec A = \frac{1}{\cos A}\), this simplifies to:

\(\frac{1}{\cos A} \times \frac{\cos A}{\sin A} = \frac{1}{\sin A}\)

This is the reciprocal of \(\sin A\), which is \(\csc A\).

Thus, the simplest form of the given expression is:

\(\csc A\)

Therefore, the correct answer is \(\csc A\).

Let's rule out other options:

• \(\sin A\): Incorrect, as shown, it simplifies to \(\csc A\).
• \(\tan A\): Incorrect, because the simplification process demonstrates this doesn't appear as the final form.
• \(\cos A\): Incorrect, it's not equivalent to the expression after simplification.

Using trigonometric identities and properties, we have verified that the correct answer is indeed \(\csc A\).