Question

How do you find the exact functional value sin 110° cos 20° - cos 110° sin 20° using the cosine sum or difference identity?

Answer 1

The expression evaluates to ' 1 '.

Explanation:

Using the formula for sine of difference of two angles that is,
sin (A-B) = (sin A cos B - cos A sin B), the expression can very easily be evaluated. In your problem A=110° and B=20°.
therefore sin(A-B) = sin(110°-20°)= sin (90°)=1

But if you insist on using the cosine sum or difference identity, you can first convert the expression to match with the formula of cosine sum or difference.You can proceed in the following way:

= sin(110°) cos(20°) - cos(110°) sin(20°)
= sin(90°+20°) cos (20°) - cos (90°+20°) sin (20°)

Here we can use the formulas like
sin ( 90°+x)= cos(x) and cos (90°+x) = {-sin(x)}

= sin(20°) cos (20°) - {-sin(20°)} sin (20°)
= cos(20°) cos (20°) + sin(20°) sin (20°)

And we know,
cos(A-B) = cos A cos B + sin A sin B

= cos (20°-20°)
= cos 0°
= 1