Question

If \[\sin \theta -\cos \theta =0\]then the value is \[({{\sin }^{4}}\theta +{{\cos }^{4}}\theta )\]
A. 1
B. \[\dfrac{3}{4}\]
C. \[\dfrac{1}{2}\]
D. \[\dfrac{1}{2}\]

Answer 1

Hint: We can solve this question by finding the value of $\theta $ from the given condition and then we can find the value of the required expression at a particular value of $\theta $.
Complete step-by-step answer:
\[\sin \theta -\cos \theta =0\]
\[\sin \theta =\cos \theta \] ...........................................(i)
Now we can use $\cos (\theta )=\sin \left( {{90}^{\circ }}-\theta \right)$
So we can write equation (i) as
\[\sin \theta =\sin ({{90}^{\circ }}-\theta )\]
On comparing
$\Rightarrow \theta ={{90}^{\circ }}-\theta $
$\Rightarrow \theta +\theta ={{90}^{\circ }}$
$\Rightarrow 2\theta ={{90}^{\circ }}$
$\Rightarrow \theta =\dfrac{{{90}^{\circ }}}{2}$
$\Rightarrow \theta ={{45}^{\circ }}$
Given expression is
\[\Rightarrow {{\sin }^{4}}\theta +{{\cos }^{4}}\theta \]
At $\theta ={{45}^{\circ }}$
\[\Rightarrow {{\left( \sin {{45}^{\circ }} \right)}^{4}}+{{\left( \cos {{45}^{\circ }} \right)}^{4}}\]
\[\Rightarrow {{\left( \dfrac{1}{\sqrt{2}} \right)}^{4}}+{{\left( \dfrac{1}{\sqrt{2}} \right)}^{4}}\] \[\left\{ \because \sin 45{}^\circ =\dfrac{1}{\sqrt{2}},\cos 45{}^\circ =\dfrac{1}{\sqrt{2}} \right\}\]
\[\Rightarrow \dfrac{1}{4}+\dfrac{1}{4}\]
$\Rightarrow \dfrac{1}{2}$
Hence option C is correct.
Note: In this type of question we need to be careful about choosing the value of unknown angles. In the given question there is no range of $\theta $ given. So we choose an angle in the first quadrant. But if there is a range we need to choose the value of angle according to that range.
We can solve this question by using $\tan (\theta )=\dfrac{\sin (\theta )}{\cos (\theta )}$
\[\Rightarrow \sin \theta -\cos \theta =0\]
\[\Rightarrow \sin \theta =\cos \theta \]
\[\Rightarrow \dfrac{\sin \theta }{\cos \theta }=1\]
\[\Rightarrow \tan \theta =1\]
.\[\Rightarrow \tan \theta =\tan 45{}^\circ \]
\[\Rightarrow \theta =45{}^\circ \]
Now we can substitute the value of $\theta $ and calculate the value of the given expression.