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If [tex]\sin \theta = \frac{12}{13}[/tex] and [tex]\cos \theta = \frac{5}{13}[/tex], find the value of [tex]\sec^2 \theta - \tan^2 \theta[/tex].

Sagot :

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].