If tan2 45° – cos230° = x sin 45° cos 45° then x = ?
(a) 2
(b) –2
(c) $\frac{1}{2}$
(d) $-\frac{1}{2}$
As we know that,
$\cos 45^{\circ}=\frac{1}{\sqrt{2}}$
$\cos 30^{\circ}=\frac{\sqrt{3}}{2}$
$\tan 45^{\circ}=1$
$\sin 45^{\circ}=\frac{1}{\sqrt{2}}$
By substituting these values, we get
$\tan ^{2} 45^{\circ}-\cos ^{2} 30^{\circ}=x \sin 45^{\circ} \cos 45^{\circ}$
$\Rightarrow(1)^{2}-\left(\frac{\sqrt{3}}{2}\right)^{2}=x\left(\frac{1}{\sqrt{2}}\right)\left(\frac{1}{\sqrt{2}}\right)$
$\Rightarrow 1-\frac{3}{4}=\frac{x}{2}$
$\Rightarrow \frac{4-3}{4}=\frac{x}{2}$
$\Rightarrow \frac{x}{2}=\frac{1}{4}$
$\Rightarrow x=\frac{1}{2}$
Hence, the correct option is (c).
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