Question

If sin θ – cos θ = 0 then the value of (sin⁴θ + cos⁴θ) is
(a) $\frac{1}{4}$

(b) $\frac{1}{2}$

(c) $\frac{3}{4}$

(d) 1

Answer 1

$$\begin{gathered} \sin \theta -\cos \theta =0 \\ \Rightarrow \sin \theta =\cos \theta \\ \Rightarrow \frac{\sin \theta }{\cos \theta }=1 \\ \Rightarrow \tan \theta =\tan 45° \end{gathered}$$
$$\begin{gathered} \Rightarrow \theta =45° \\ \therefore {\sin }^4\theta +{\cos }^4\theta \\ ={\sin }^{4}45°+{\cos }^{4}45° \\ ={\left(\frac{1}{\sqrt{2}}\right)}^4+{\left(\frac{1}{\sqrt{2}}\right)}^4 \\ =\frac{1}{4}+\frac{1}{4} \end{gathered}$$
$$\begin{gathered} =\frac{2}{4} \\ =\frac{1}{2} \end{gathered}$$

Hence, the correct answer is option (b).