Question

# What is the value of $({\cos ^2}{67^ \circ }{\text{ - }}{\sin ^2}{23^ \circ })?$

Hint: Here convert $\cos \theta$ into $sin \theta$ by using the trigonometric relation ${\cos ^2}({90^ \circ }{\text{ - }}\theta ){\text{ = }}{\sin ^2}\theta$.

Complete step by step solution:

Given equation is $({\cos ^2}{67^ \circ }{\text{ - }}{\sin ^2}{23^ \circ })$

Now, we can write this equation in this form i.e.

$({\cos ^2}{67^ \circ }{\text{ - }}{\sin ^2}{23^ \circ }) = {\cos ^2}({90^ \circ }{\text{ - 2}}{{\text{3}}^ \circ }{\text{) - }}{\sin ^2}{23^ \circ }$......................................(1)

We know that

${\cos ^2}({90^ \circ }{\text{ - }}\theta ){\text{ = }}{\sin ^2}\theta$

Now using this concept we can write the equation (1) in this form i.e.

$({\cos ^2}{67^ \circ }{\text{ - }}{\sin ^2}{23^ \circ }){\text{ = }}{\sin ^2}{23^ \circ }{\text{ - }}{\sin ^2}{23^ \circ }$

$({\cos ^2}{67^ \circ }{\text{ - }}{\sin ^2}{23^ \circ }) = {\text{ 0}}$

Note: - These types of problems can be solved by converting either $\cos \theta$ into $\sin \theta$ or $sin \theta$ into $\cos \theta$. Here we have converted $\cos \theta$ into $\sin \theta$ in the similar way we can do it for other questions as well.