Question

$\mathrm{If}\sin \theta =\cos \left(\theta -45°\right),\mathrm{where}\theta \mathrm{is}\mathrm{acute},\mathrm{then}\mathrm{find}\mathrm{the}\mathrm{value}\mathrm{of}\theta .$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \sin \theta =\cos \left(\theta -45°\right) \\ \Rightarrow \cos \left(90°-\theta \right)=\cos \left(\theta -45°\right) \\ \mathrm{Comparing}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ 90°-\theta =\theta -45° \\ \Rightarrow \theta +\theta =90°+45° \\ \Rightarrow 2\theta =135° \\ \Rightarrow \theta ={\left(\frac{135}{2}\right)}^{°} \\ \therefore \theta =67.5° \end{gathered}$$