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8. Introduction to Trigonometry
medium
Write 'True' or 'False' and justify your answer.
The value of the expression $\left(\cos ^{2} 23^{\circ}-\sin ^{2} 67^{\circ}\right)$ is positive.
Option A
Option B
Option C
Option D
Solution
False
$\cos ^{2} 23^{\circ}-\sin ^{2} 67^{\circ}=\left(\cos 23^{\circ}-\sin 67^{\circ}\right)\left(\cos 23^{\circ}+\sin 67^{\circ}\right)\left[\because\left(a^{2}-b^{2}\right)=(a-b)(a+b)\right]$
$=\left[\cos 23^{\circ}-\sin \left(90^{\circ}-23^{\circ}\right)\right]\left(\cos 23^{\circ}+\sin 67^{\circ}\right)$
$=\left(\cos 23^{\circ}-\cos 23^{\circ}\right)\left(\cos 23^{\circ}+\sin 67^{\circ}\right)\left[\because \sin \left(90^{\circ}-0\right)=\cos 0\right]$
$=0 \cdot\left(\cos 23^{\circ}+\sin 67^{\circ}\right)=0$
which may be either positive or negative.
Standard 10
Mathematics
