Question

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

Answer 1

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.