Question

If $\sin \left(A\right)-\cos \left(A\right)=0$, then find the value of ${\sin }^4\left(A\right)+{\cos }^4\left(A\right)$

• A. $0$
• B. $1$
• C. $\frac{1}{2}$
• D. $\frac{3}{4}$

Answer 1

The correct option is C

$\frac{1}{2}$

Explanation for the correct option:

Step 1: Find the value of the angle $A$.

Given: $\sin \left(A\right)-\cos \left(A\right)=0$

$$\begin{gathered} \Rightarrow \sin \left(A\right)=\cos \left(A\right) \\ \Rightarrow A=45° \\ \Rightarrow \sin \left(A\right)=\cos \left(A\right)=\frac{1}{\sqrt{2}} \end{gathered}$$

Step 2: Find the value of the given expression.

Since, $\sin \left(A\right)=\cos \left(A\right)=\frac{1}{\sqrt{2}}$.

So,

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

Therefore, the value of the given expression is $\frac{1}{2}$.

Hence, option (C) is the correct option.