Question

Determine whether the following equation is identity or not:

${\sin }^4\mathrm{\theta }+{\cos }^4\mathrm{\theta }=1-2{\sin }^2\mathrm{\theta }{\cos }^2\mathrm{\theta }$.

Answer 1

Verify L.H.S and R.H.S at any value of $\mathit{\theta }$

${\sin }^4\mathrm{\theta }+{\cos }^4\mathrm{\theta }=1-2{\sin }^2\mathrm{\theta }{\cos }^2\mathrm{\theta }$

Let, $\mathrm{L}.\mathrm{H}.\mathrm{S}.={\sin }^4\mathrm{\theta }+{\cos }^4\mathrm{\theta }$

when $\mathrm{\theta }={30}^{\circ }$

$\mathrm{L}.\mathrm{H}.\mathrm{S}.={\sin }^4{30}^{\circ }+{\cos }^4{30}^{\circ }$

$$\begin{gathered} ={\left(\frac{1}{2}\right)}^4+{\left(\frac{\sqrt{3}}{2}\right)}^4 \\ =\frac{1}{16}+\frac{9}{16} \\ =\frac{10}{16} \\ =\frac{5}{8} \end{gathered}$$

$\mathrm{R}.\mathrm{H}.\mathrm{S}.=1-2{\sin }^2{30}^{\circ }{\cos }^2{30}^{\circ }$

$$\begin{gathered} =1-2{\left(\frac{1}{2}\right)}^2\times {\left(\frac{\sqrt{3}}{2}\right)}^2 \\ =1-2\times \frac{1}{4}\times \frac{3}{4} \\ =1-\frac{3}{8} \\ =\frac{5}{8} \end{gathered}$$

$\mathrm{L}.\mathrm{H}.\mathrm{S}.=\mathrm{R}.\mathrm{H}.\mathrm{S}.$

Hence, the given equation is an identity.