Question

How is this solved using trig identities (sum/difference)?

Answer 1

The angle sum identity for cosine is:

$cos( \theta+ \beta) = cos( \theta)cos( \beta) - sin( \theta)sin( \beta)$

So we need to figure out the values of \theta and \beta.

$tan^{-1}( \frac{1}{2}) = \theta$

$cos^{-1}( \frac{2}{3} ) = \beta$

$cos(tan^{-1}( \frac{1}{2}))cos(cos^{-1}(\frac{2}{3})) - sin(tan^{-1}( \frac{1}{2}))sin(cos^{-1}(\frac{2}{3})) =$ 0.26295 rad