Find the Exact Value

Note that   sin( 90° - a )  =  cos( a )      and      cos( 90° - a )  =  sin( a )      and      tan( 90° - a )  =  cot( a )

This might help you see why those are true....

sin( 90° - a )  =  cos( a )   is saying that line  r  is the same length as line  s .

cos( 90° - a )  =  sin( a )   is saying that line  b  is the same length as line  c .

tan( 90° - a )  =  cot( a )   is saying that   r / b  =  s / c

So.....

     sin( 38° )  -  cos( 52° )

=   sin( 90° - 52° )  -  cos( 52° )

=   cos( 52° )  -  cos( 52° )

=   0

     tan( 12° )  -  cot( 78° )

=   tan( 90° - 78° )  -  cot( 78° )

...Do you see where this one is going? Can you finish it?

$\phantom{=}\qquad\frac{\cos(10°)}{\sin(80°)}\\~\\ =\qquad\frac{\cos(90° - 80°)}{\sin(80°)}\\~\\ =\qquad\frac{\sin(80°)}{\sin(80°)}\\~\\ =\qquad1$

The last one is a really similar to previous one. Do you know how to do it now?