Question

Evaluate \[ \sec 2\theta - \tan 2\theta = \]

• A. \(\tan\!\left(\dfrac{\pi}{4}-\theta\right)\)
• B. \(\tan 2\theta\)
• C. \(\cot 2\theta\)
• D. \(\cot\!\left(\dfrac{\pi}{4}-\theta\right)\)

Hint:

Remember that \(\sec x - \tan x\) can be simplified using half–angle identities.

Answer 1

The Correct Option is A

Step 1: Use standard trigonometric identity.
We know that \[ \sec x - \tan x = \tan\!\left(\frac{\pi}{4} - \frac{x}{2}\right) \]
Step 2: Substitute \(x = 2\theta\).
\[ \sec 2\theta - \tan 2\theta = \tan\!\left(\frac{\pi}{4} - \theta\right) \]
Step 3: Final result.
Thus, \[ \sec 2\theta - \tan 2\theta = \tan\!\left(\frac{\pi}{4}-\theta\right) \]