Question

If we have a trigonometric expression $\sin x-\cos x=0$, then what is the value of ${{\sin }^{4}}x+{{\cos }^{4}}x$?
A). 1
B). $\dfrac{3}{4}$
C). $\dfrac{1}{2}$
D). $\dfrac{1}{4}$

Answer 1

Take the relation $\sin x-\cos x=0$ as $\sin x=\cos x$ then put this relation in the identity ${{\sin }^{2}}x+{{\cos }^{2}}x=1$ then from that get the value of ${{\sin }^{2}}x$ and ${{\cos }^{2}}x$ respectively and then further put it back in the expression ${{\sin }^{4}}x+{{\cos }^{4}}x$ to get the answer.

Complete step-by-step solution:
In the question we are given an equation of $\sin x$ and $\cos x$ which is $\sin x-\cos x=0$ and we have to find the value of ${{\sin }^{4}}x+{{\cos }^{4}}x$.
So we are given that $\sin x-\cos x=0$ or $\sin x=\cos x$. So we have to find the value such that $\sin x$ is equal to $\cos x$.
We know a universal satisfying identity for all ‘x’, ${{\sin }^{2}}x+{{\cos }^{2}}x=1$ which is applicable for all values of ‘x’. So, we are given, $\sin x=\cos x$. Now we will take $\sin x$ as $\cos x$ and replace it in the identity ${{\sin }^{2}}x+{{\cos }^{2}}x=1$ to proceed. So we are given,
${{\sin }^{2}}x+{{\cos }^{2}}x=1$
On replacing $\sin x$ by $\cos x$ we get,
${{\cos }^{2}}x+{{\cos }^{2}}x=1$
Or, $2{{\cos }^{2}}x=1$
So, the value of ${{\cos }^{2}}x=\dfrac{1}{2}$.
If $\sin x=\cos x$, then ${{\sin }^{2}}x={{\cos }^{2}}x$. So, we found out ${{\cos }^{2}}x=\dfrac{1}{2}$. So, ${{\sin }^{2}}x=\dfrac{1}{2}$.
Hence, we were asked to find ${{\sin }^{4}}x+{{\cos }^{4}}x$ which we can write as, ${{\left( {{\sin }^{2}}x \right)}^{2}}+{{\left( {{\cos }^{2}}x \right)}^{2}}$.
Now, as we know, ${{\sin }^{2}}x={{\cos }^{2}}x=\dfrac{1}{2}$.
So, we get,
${{\left( \dfrac{1}{2} \right)}^{2}}+{{\left( \dfrac{1}{2} \right)}^{2}}$ or $\dfrac{1}{4}+\dfrac{1}{4}$ or $\dfrac{1}{2}$.
Hence for given condition $\sin x-\cos x=0$ then ${{\sin }^{4}}x+{{\cos }^{4}}x$ will be $\dfrac{1}{2}$.
So, the correct option is (c).

Note: There is an another method to solve this problem by writing given $\sin x-\cos x=0$ then$\sin x=\cos x$ as $\dfrac{\sin x}{\cos x}=1$ and then $\tan x=1$ and hence we will finding the value of x. Then put the value of x in the expression ${{\sin }^{4}}x+{{\cos }^{4}}x$ then after solving this equation we will get the final answer which is $\dfrac{1}{2}$.