We are asked to evaluate: sec²60° – 1

Let's solve this step by step.

We know that secant is the reciprocal of cosine.

$\sec 60 ° = \frac{1}{\cos 60 °}$

The value of cos 60° is $\frac{1}{2}$ .

Therefore, $\sec 60 ° = \frac{1}{1 / 2} = 2$

Now we square the value we just found.

$\sec^{2} 60 ° = 2^{2} = 4$

Now we substitute this value back into the original expression.

$\sec^{2} 60 ° - 1 = 4 - 1 = 3$

Therefore, sec²60° – 1 = 3

This problem is a direct application of a fundamental Pythagorean identity:

$\sec^{2} θ - 1 = \tan^{2} θ$

If we had recognized this identity first, we could have solved it even faster:

$\sec^{2} 60 ° - 1 = \tan^{2} 60 °$

We know that tan 60° = √3, so tan²60° = (√3)² = 3.

This confirms our answer of 3.

Angle (θ)	sin θ	cos θ	tan θ	sec θ
0°	0	1	0	1
30°	1/2	√3/2	1/√3	2/√3
45°	1/√2	1/√2	1	√2
60°	√3/2	1/2	√3	2
90°	1	0	∞	∞

Pythagorean Identities:
- $\sin^{2} θ + \cos^{2} θ = 1$
- $1 + \tan^{2} θ = \sec^{2} θ$ (used in this problem)
- $1 + \cot^{2} θ = \csc^{2} θ$
Reciprocal Identities:
- $\sec θ = \frac{1}{\cos θ}$
- $\csc θ = \frac{1}{\sin θ}$
- $\cot θ = \frac{1}{\tan θ}$