Question

If $\cos \left(A\right)+{\cos }^2\left(A\right)=1$, then ${\sin }^2\left(A\right)+{\sin }^4\left(A\right)=1$. Write True or False and justify your answer

• A. True
• B. False

Answer 1

The correct option is A True

Simplify the given expression

Given that

$$\begin{gathered} \cos \left(A\right)+{\cos }^2\left(A\right)=1 \\ \Rightarrow \cos \left(A\right)=1-{\cos }^2\left(A\right)...................\left(1\right) \end{gathered}$$

We know that

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

We get

$\cos \left(A\right)={\sin }^2\left(A\right)....................\left(2\right)$

$\therefore {\cos }^2\left(A\right)={\sin }^4\left(A\right).........................\left(3\right)$

Now using equation $\left(2\right)$ and $\left(3\right)$ in equation $\left(1\right)$ we get

$$\begin{gathered} {\sin }^2\left(A\right)=1-{\sin }^4\left(A\right) \\ \Rightarrow {\sin }^2\left(A\right)+{\sin }^4\left(A\right)=1 \end{gathered}$$

Hence, ${\sin }^2\left(A\right)+{\sin }^4\left(A\right)=1$, so, the given statement is true.