Question

The value of \(\sec^2 45^\circ - \tan^2 45^\circ\) is :

• A. \(\frac{1}{\sqrt{2}}\)
• B. \(\frac{\sqrt{3}}{2}\)
• C. \(-1\)
• D. 1

Hint:

You can solve this in two ways: {1. Using values:} \(\sec 45^\circ = \sqrt{2}\) \(\tan 45^\circ = 1\) So, \(\sec^2 45^\circ - \tan^2 45^\circ = (\sqrt{2})^2 - (1)^2 = 2 - 1 = 1\). {2. Using identity:} Recall the Pythagorean identity: \(1 + \tan^2\theta = \sec^2\theta\). Rearrange it: \(\sec^2\theta - \tan^2\theta = 1\). This is true for any valid angle \(\theta\), including \(45^\circ\). So the answer is directly 1.

Answer 1

The Correct Option is D

Concept: This problem can be solved in two ways: 1. By substituting the known trigonometric values for \(45^\circ\). 2. By using the Pythagorean trigonometric identity \(1 + \tan^2\theta = \sec^2\theta\). Method 1: Substituting trigonometric values Step 1: Recall the values of \(\sec 45^\circ\) and \(\tan 45^\circ\)
We know \(\cos 45^\circ = \frac{1}{\sqrt{2}}\). Since \(\sec \theta = \frac{1}{\cos \theta}\), then \(\sec 45^\circ = \frac{1}{1/\sqrt{2}} = \sqrt{2}\).
\(\tan 45^\circ = 1\). Step 2: Calculate \(\sec^2 45^\circ\) and \(\tan^2 45^\circ\)
\(\sec^2 45^\circ = (\sec 45^\circ)^2 = (\sqrt{2})^2 = 2\).
\(\tan^2 45^\circ = (\tan 45^\circ)^2 = (1)^2 = 1\). Step 3: Calculate the expression \(\sec^2 45^\circ - \tan^2 45^\circ\) \[ \sec^2 45^\circ - \tan^2 45^\circ = 2 - 1 = 1 \] Method 2: Using the Pythagorean Identity Step 1: Recall the Pythagorean identity involving secant and tangent One of the fundamental Pythagorean trigonometric identities is: \[ 1 + \tan^2\theta = \sec^2\theta \] Step 2: Rearrange the identity Subtract \(\tan^2\theta\) from both sides of the identity: \[ \sec^2\theta - \tan^2\theta = 1 \] This identity holds true for any angle \(\theta\) for which \(\sec\theta\) and \(\tan\theta\) are defined (i.e., \(\theta\) is not an odd multiple of \(90^\circ\)). Step 3: Apply the identity for \(\theta = 45^\circ\) For \(\theta = 45^\circ\), the identity directly gives: \[ \sec^2 45^\circ - \tan^2 45^\circ = 1 \] Both methods yield the value 1. This matches option (4).