Question

If tan² 45° – cos²30° = x sin 45° cos 45° then x = ?

(a) 2

(b) –2

(c) $\frac{1}{2}$

(d) $-\frac{1}{2}$

Answer 1

As we know that,
​$$\begin{gathered} \cos 45°=\frac{1}{\sqrt{2}} \\ \cos 30°=\frac{\sqrt{3}}{2} \\ \tan 45°=1 \\ \sin 45°=\frac{1}{\sqrt{2}} \\ \mathrm{By}\mathrm{substituting}\mathrm{these}\mathrm{values},\mathrm{we}\mathrm{get} \\ {\tan }^{2}45°-{\cos }^{2}30°=x\sin 45°\cos 45° \\ \Rightarrow {\left(1\right)}^2-{\left(\frac{\sqrt{3}}{2}\right)}^2=x\left(\frac{1}{\sqrt{2}}\right)\left(\frac{1}{\sqrt{2}}\right) \\ \Rightarrow 1-\frac{3}{4}=\frac{x}{2} \\ \Rightarrow \frac{4-3}{4}=\frac{x}{2} \\ \Rightarrow \frac{x}{2}=\frac{1}{4} \\ \Rightarrow x=\frac{1}{2} \\ \mathrm{Hence},\mathrm{the}\mathrm{correct}\mathrm{option}\mathrm{is}\left(\mathrm{c}\right). \end{gathered}$$