Question

Question 2
Write ‘True’ or ‘False’ and justify your answer in each of the following:
The value of the expression $(co{s}^2{23}^{\circ }-si{n}^2{67}^{\circ })$ is positive.

Answer 1

The statement is false.

As, ${67}^{\circ }and{23}^{\circ }$ are complementary angles,

so,
$co{s}^2{23}^{\circ }-si{n}^2{67}^{\circ }=(cos{23}^{\circ }-sin{67}^{\circ })(cos{23}^{\circ }+sin{67}^{\circ })$

$\lfloor \because (a^2-b^2)=(a-b)(a+b)\rfloor$

$=\lfloor cos{23}^{\circ }-sin({90}^{\circ }-{23}^{\circ })\rfloor (cos{23}^{\circ }+sin{67}^{\circ })$

$=(cos{23}^{\circ }-cos{23}^{\circ })(cos{23}^{\circ }+sin{67}^{\circ })\lfloor \because sin({90}^{\circ }-\theta )=cos\theta \rfloor$

$=0$