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Write 'True' or 'False' and justify your answer in each of the following:
The value of the expression \( \left(\cos ^{2} 23^{\circ}-\sin ^{2} 67^{\circ}\right) \) is positive.
Given:
The value of the expression \( \left(\cos ^{2} 23^{\circ}-\sin ^{2} 67^{\circ}\right) \) is positive.
To do:
We have to find whether the given statement is true or false.
Solution:
$\cos ^{2} 23^{\circ}-\sin ^{2} 67^{\circ}=(\cos 23^{\circ}-\sin 67^{\circ})(\cos 23^{\circ}+\sin 67^{\circ})$ [Since $(a^{2}-b^{2})=(a-b)(a+b)$]
$=[\cos 23^{\circ}-\sin(90^{\circ}-23^{\circ})](\cos 23^{\circ}+\sin 67^{\circ})$
$=(\cos 23^{\circ}-\cos 23^{\circ})(\cos 23^{\circ}+\sin 67^{\circ})$ [Since $\sin (90^{\circ}-\theta)=\cos \theta]$
$=0 \times (\cos 23^{\circ}+\sin 67^{\circ})$
$=0$
which is neither positive nor negative.
The given statement is false.