Question

How do you prove `cos(u-v)/(cosusinv)=tanu+cotv`?

Answer 1

see explanation

Explanation:

Attempt to convert the left side into the form of the right side.

Consider the numerator of the function on the left. Using the appropriate `color(blue)"addition formula"`

`color(orange)"Reminder " color(red)(|bar(ul(color(white)(a/a)color(black)(cos(A-B)=cosAcosB+sinAsinB)color(white)(a/a)|)))`

`rArrcos(u-v)=cosucosv+sinusinv`

We now have : `(cosucosv+sinusinv)/(cosusinv)`

now divide the terms on the numerator by the denominator.

`rArr(cancel(cosu)cosv)/(cancel(cosu)sinv)+(sinucancel(sinv))/(cosucancel(sinv))=(cosv)/(sinv)+(sinu)/(cosu)`

`color(orange)"Reminder " color(red)(|bar(ul(color(white)(a/a)color(black)(tantheta=(sintheta)/(costheta)" and " cottheta=(costheta)/(sintheta))color(white)(a/a)|)))`

`rArr(cosv)/(sinv)+(sinu)/(cosu)=cotv+tanu=tanu+cotv`

Thus left side = right side `rArr" proven"`