State whether the following is true or false. Justify your answer.
$\sin \theta = \cos \theta$ for all values of $\theta$.
$\sin \theta = \cos \theta$ for all values of $\theta$.
(B) The statement $\sin \theta = \cos \theta$ for all values of $\theta$ is false.
Justification:
$1$. The equation $\sin \theta = \cos \theta$ holds true only when $\theta = 45^{\circ}$.
$2$. At $\theta = 45^{\circ}$,$\sin 45^{\circ} = \frac{1}{\sqrt{2}}$ and $\cos 45^{\circ} = \frac{1}{\sqrt{2}}$.
$3$. For other values of $\theta$,the statement is incorrect. For example,at $\theta = 30^{\circ}$,$\sin 30^{\circ} = \frac{1}{2}$ while $\cos 30^{\circ} = \frac{\sqrt{3}}{2}$.
$4$. Since $\frac{1}{2} \neq \frac{\sqrt{3}}{2}$,the statement is false.
Justification:
$1$. The equation $\sin \theta = \cos \theta$ holds true only when $\theta = 45^{\circ}$.
$2$. At $\theta = 45^{\circ}$,$\sin 45^{\circ} = \frac{1}{\sqrt{2}}$ and $\cos 45^{\circ} = \frac{1}{\sqrt{2}}$.
$3$. For other values of $\theta$,the statement is incorrect. For example,at $\theta = 30^{\circ}$,$\sin 30^{\circ} = \frac{1}{2}$ while $\cos 30^{\circ} = \frac{\sqrt{3}}{2}$.
$4$. Since $\frac{1}{2} \neq \frac{\sqrt{3}}{2}$,the statement is false.
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$(i)$ $\cos A$ is the abbreviation used for the cosecant of angle $A$.
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