If Cos \(\theta\) = \(\frac{12}{13}\). Find 1 + cot\(^2\)\(\theta\)

If Cos \(\theta\) = \(\frac{12}{13}\). Find 1 + cot\(^2\)\(\theta\)

Mathematics

JAMB 1990

If Cos \(\theta\) = \(\frac{12}{13}\). Find 1 + cot\(^2\)\(\theta\)

A. \(\frac{169}{25}\)
B. \(\frac{25}{169}\)
C. \(\frac{169}{144}\)
D. \(\frac{144}{169}\)

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Correct Answer: Option A

Explanation

Cos \(\theta\) = \(\frac{12}{13}\)

x\(^2\) + 12\(^2\) = 13\(^2\)

x\(^2\) = 169- 144 = 25

x = \(\sqrt{25}\) = 5

Hence, tan\(\theta\) = \(\frac{5}{12}\) and cos\(\theta\) = \(\frac{12}{13}\)

If cot\(^2\)\(\theta\) = 1 + \(\frac{1}{tan^2\theta}\)

= 1 + \(\frac{1}{\frac{(5)^2}{12^2}}\)

= 1 + \(\frac{1}{\frac{25}{144}}\)

= 1 + \(\frac{144}{25}\)

= \(\frac{25 + 144}{25}\)

= \(\frac{169}{25}\)