Question

The value of ${\cos }^4\mathrm{\theta }+{\sin }^4\mathrm{\theta }-6{\cos }^2\mathrm{\theta }{\sin }^2\mathrm{\theta }$
(a) $\cos 2\mathrm{\theta }$
(b) $\sin 2\mathrm{\theta }$
(c) $\cos 4\mathrm{\theta }$
(d) none of these

Answer 1

(c) $\cos 4\mathrm{\theta }$

$$\begin{gathered} {\cos }^4\mathrm{\theta }+{\sin }^4\mathrm{\theta }-6{\cos }^2\mathrm{\theta }{\sin }^2\mathrm{\theta }={\cos }^4\mathrm{\theta }+{\sin }^4\mathrm{\theta }-2{\cos }^2\mathrm{\theta }{\sin }^2\mathrm{\theta }-4{\cos }^2\mathrm{\theta }{\sin }^2\mathrm{\theta } \\ ={\left({\cos }^2\mathrm{\theta }-{\sin }^2\mathrm{\theta }\right)}^2-{\left(2\sin \mathrm{\theta }\cos \mathrm{\theta }\right)}^2 \\ ={\cos }^{2}2\mathrm{\theta }-{\sin }^{2}2\mathrm{\theta } \\ =\cos 4\mathrm{\theta } \end{gathered}$$