What is the value of cos(arctan(4/3)+arccos(5/13)) ?

The value of cos(arctan(4/3)+arccos(5/13)) is -33/65

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cos(arctan(4/3)+arccos(5/13))

cos (arctan (\frac{4}{3}) + arccos (\frac{5}{13}))

- \frac{33}{65}

Solution steps

cos (arctan (\frac{4}{3}) + arccos (\frac{5}{13}))

Rewrite using trig identities:

cos (arctan (\frac{4}{3})) cos (arccos (\frac{5}{13})) - sin (arctan (\frac{4}{3})) sin (arccos (\frac{5}{13}))

= cos (arctan (\frac{4}{3})) cos (arccos (\frac{5}{13})) - sin (arctan (\frac{4}{3})) sin (arccos (\frac{5}{13}))

Rewrite using trig identities:

cos (arctan (\frac{4}{3})) = \frac{3}{5}

Rewrite using trig identities:

cos (arccos (\frac{5}{13})) = \frac{5}{13}

Rewrite using trig identities:

sin (arctan (\frac{4}{3})) = \frac{4}{5}

Rewrite using trig identities:

sin (arccos (\frac{5}{13})) = \frac{12}{13}

= \frac{3}{5} \cdot \frac{5}{13} - \frac{4}{5} \cdot \frac{12}{13}

Simplify

\frac{3}{5} \cdot \frac{5}{13} - \frac{4}{5} \cdot \frac{12}{13} : - \frac{33}{65}

= - \frac{33}{65}

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What is the value of cos(arctan(4/3)+arccos(5/13)) ?
The value of cos(arctan(4/3)+arccos(5/13)) is -33/65