Question

If $\cos \left(x\right)+{\cos }^2\left(x\right)=1$. then prove that ${\sin }^2\left(x\right)+{\sin }^4\left(x\right)=1$

Answer 1

Prove the given expression

It is given that

$$\begin{gathered} \cos \left(x\right)+{\cos }^2\left(x\right)=1 \\ \Rightarrow \cos \left(x\right)=1-{\cos }^2\left(x\right) \\ \Rightarrow \cos \left(x\right)={\sin }^2\left(x\right)\left[\because {\sin }^2\left(x\right)+{\cos }^2\left(x\right)=1\right] \\ \Rightarrow {\left(\cos \left(x\right)\right)}^2={\left({\sin }^2\left(x\right)\right)}^2\left[\mathrm{Squaring}\mathrm{on}\mathrm{both}\mathrm{side}\right] \\ \Rightarrow {\cos }^2\left(x\right)={\sin }^4\left(x\right) \\ \Rightarrow 1-{\sin }^2\left(x\right)={\sin }^4\left(x\right)\left[\because {\sin }^2\left(x\right)+{\cos }^2\left(x\right)=1\right] \\ \Rightarrow {\sin }^4\left(x\right)+{\sin }^2\left(x\right)=1 \end{gathered}$$

Hence Proved.