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Hint: Use the formula $\sin \theta = \cos ({90^ \circ } - \theta )$ and then equate angles on both sides.

Complete step-by-step answer:

From the question,

$ \Rightarrow \sin \theta = \cos (\theta - 45){\text{ }}.....(i)$

We know that, $\sin \theta = \cos (90 - \theta ),$putting this value in equation $(i)$:

\[

\Rightarrow \cos ({90^ \circ } - \theta ) = \cos (\theta - {45^ \circ }), \\

\Rightarrow {90^ \circ } - \theta = \theta - {45^ \circ }, \\

\Rightarrow 2\theta = {135^ \circ }, \\

\Rightarrow \theta = \dfrac{{{{135}^ \circ }}}{2} = 67{\dfrac{1}{2}^ \circ } \\

\]

Therefore, the value of angle $\theta $ is \[67{\dfrac{1}{2}^ \circ }.\]

Note: Whenever we need to solve a trigonometric equation, we try to convert both sides of the equation in the same trigonometric ratio so that we can easily compare their angles.

Complete step-by-step answer:

From the question,

$ \Rightarrow \sin \theta = \cos (\theta - 45){\text{ }}.....(i)$

We know that, $\sin \theta = \cos (90 - \theta ),$putting this value in equation $(i)$:

\[

\Rightarrow \cos ({90^ \circ } - \theta ) = \cos (\theta - {45^ \circ }), \\

\Rightarrow {90^ \circ } - \theta = \theta - {45^ \circ }, \\

\Rightarrow 2\theta = {135^ \circ }, \\

\Rightarrow \theta = \dfrac{{{{135}^ \circ }}}{2} = 67{\dfrac{1}{2}^ \circ } \\

\]

Therefore, the value of angle $\theta $ is \[67{\dfrac{1}{2}^ \circ }.\]

Note: Whenever we need to solve a trigonometric equation, we try to convert both sides of the equation in the same trigonometric ratio so that we can easily compare their angles.

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