Here's a step-by-step solution to find the value of [tex]\( \sec^2 \theta - \tan^2 \theta \)[/tex] given [tex]\( \sin \theta = \frac{12}{13} \)[/tex] and [tex]\( \cos \theta = \frac{5}{13} \)[/tex]:
1. Compute [tex]\(\sec \theta\)[/tex]:
[tex]\[
\sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{5}{13}} = \frac{13}{5}
\][/tex]
2. Calculate [tex]\(\sec^2 \theta\)[/tex]:
[tex]\[
\sec^2 \theta = \left( \frac{13}{5} \right)^2 = \frac{169}{25} = 6.76
\][/tex]
3. Compute [tex]\(\tan \theta\)[/tex]:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{12}{13}}{\frac{5}{13}} = \frac{12}{5}
\][/tex]
4. Calculate [tex]\(\tan^2 \theta\)[/tex]:
[tex]\[
\tan^2 \theta = \left( \frac{12}{5} \right)^2 = \frac{144}{25} = 5.76
\][/tex]
5. Find the value of [tex]\(\sec^2 \theta - \tan^2 \theta\)[/tex]:
[tex]\[
\sec^2 \theta - \tan^2 \theta = 6.76 - 5.76 = 1
\][/tex]
Therefore, the value of [tex]\( \sec^2 \theta - \tan^2 \theta \)[/tex] is [tex]\( 1 \)[/tex].
