[tex] \huge \mathbb{\green{\fbox{\pink{ANSWER}}}} [/tex]
[tex] \\ \: \: \implies \: sin^{4} \: \theta + cos^{4} \: \theta \rightarrow \blue{ \frac{1}{2} } [/tex]
[tex] \\ \\ \: \large \mathfrak \green{Solution} : [/tex]
[tex] \\ \bf \: Given : [/tex]
[tex] \\
\sf \: \: \bf sin \: \theta - cos \: \theta = 0 [/tex]
[tex] \\ \sf \rightarrow \: sin \: \theta = cos \: \theta [/tex]
[tex] \\ \sf \rightarrow \: \frac{sin \: \theta} {cos \: \theta} = 1 [/tex]
[tex] \\ \sf \rightarrow \: tan \: \theta = 1 [/tex]
[tex] \\ \sf \rightarrow \: tan \: \theta = tan \: 45 \degree [/tex]
[tex] \\ \sf \rightarrow \: \theta = 45 \degree [/tex]
[tex] \\ \\ \therefore \: sin^{4} \: \theta + cos^{4} \: \theta [/tex]
[tex] \\ \sf \rightarrow \: sin^{4} \: 45 \degree + cos^{4} \: 45 \degree [/tex]
[tex] \\ \\
\sf \rightarrow \: (\frac{1}{ \sqrt{2} } ) ^{4} +(\frac{1}{ \sqrt{2} } ) ^{4} [/tex]
[tex] \\ \sf \rightarrow \: ( \frac{1}{ \cancel {\sqrt{2} }} )^{ \cancel{2}} ( \frac{1}{ \cancel{ \sqrt{2} }} )^{\cancel{2}} + ( \frac{1}{ \cancel {\sqrt{2} }} )^{ \cancel{2}} ( \frac{1}{ \cancel{ \sqrt{2} }} )^{\cancel{2}} [/tex]
[tex] \\ \\
\sf \rightarrow \: \frac{1}{2} × \frac{1}{2} + \frac{1}{2}× \frac{1}{2} [/tex]
[tex] \\ \\ \sf \rightarrow \: \frac{1}{4} + \frac{1}{4} [/tex]
[tex] \\ \\ \sf \rightarrow \: {\cancel \frac{2} {4}}[/tex]
[tex] \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \implies \fbox{\pink {\frac{1}{2}}} [/tex]
