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Prove that:cos^(-1)(4/5)+cos^(-1)(12/13)...

Prove that:`cos^(-1)(4/5)+cos^(-1)(12/13)=cos^(-1)(33/65)`

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To prove that \( \cos^{-1}\left(\frac{4}{5}\right) + \cos^{-1}\left(\frac{12}{13}\right) = \cos^{-1}\left(\frac{33}{65}\right) \), we will use the relationship between inverse cosine and inverse tangent. ### Step-by-Step Solution: 1. **Convert \( \cos^{-1}\left(\frac{4}{5}\right) \) to \( \tan^{-1} \)**: - Let \( \theta_1 = \cos^{-1}\left(\frac{4}{5}\right) \). - By definition, \( \cos(\theta_1) = \frac{4}{5} \). - We can find the opposite side (perpendicular) using the Pythagorean theorem: ...
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