Question

If $\mathrm{sin\theta }+{\sin }^2\mathrm{\theta }=1$, then prove that ${\cos }^2\mathrm{\theta }+{\cos }^4\mathrm{\theta }=1$.

Answer 1

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

$\mathrm{sin\theta }+{\sin }^2\mathrm{\theta }=1—\left(\mathrm{i}\right)$

We have to prove that ${\cos }^2\mathrm{\theta }+{\cos }^4\mathrm{\theta }=1$

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

$\mathrm{L}.\mathrm{H}.\mathrm{S}.={\cos }^2\mathrm{\theta }+{\cos }^4\mathrm{\theta }$

$$\begin{gathered} ={\cos }^2\mathrm{\theta }+{\left({\cos }^2\mathrm{\theta }\right)}^2 \\ =1-{\sin }^2\mathrm{\theta }+{(1-{\sin }^2\mathrm{\theta })}^2 \end{gathered}$$

Equation (i) put in L.H.S.

$$\begin{gathered} =1-{\sin }^2\mathrm{\theta }+{\sin }^2\mathrm{\theta } \\ =1 \end{gathered}$$

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