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.

AcademicMathematicsNCERTClass 10

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.

raja
Updated on 10-Oct-2022 13:28:58

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