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Question 1 )

If sin θ – cos θ = 0, find the value of sin4 θ + cos4 θ. (2012, 2017D).

Notes ;

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[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]

The value of [tex]sin^4\theta+cos^4 \theta[/tex] will be [tex]\frac{2}{(\sqrt{2})^4}[/tex]

Given the expression

sin θ – cos θ = 0

This can be simplified further as:

sin θ = cos θ

sin θ/cosθ = cosθ/cosθ
tanθ = 1

θ = arctan(1)

θ = 45 degrees

To evaluate [tex]sin^4\theta+cos^4 \theta[/tex], substitute the value of theta into the expression to have:

[tex]sin^4\theta+cos^4 \theta\\=sin^445+cos^4 45\\=(\frac{1}{\sqrt{2}} )^4+(\frac{1}{\sqrt{2}} )^4\\=2(\frac{1}{(\sqrt{2})^4} )[/tex]

Hence the value of [tex]sin^4\theta+cos^4 \theta[/tex] will be [tex]\frac{2}{(\sqrt{2})^4}[/tex]

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