Question

If ${\text{sin}}\theta {\text{ + cos}}\theta {\text{ = }}\sqrt 2 \cos \left( {90^\circ - \theta } \right)$, then find the value of ${\text{cot}}\theta$.
${\text{A}}{\text{. }}\frac{1}{2} \\\ {\text{B}}{\text{. 0}} \\\ {\text{C}}{\text{. }}\sqrt 2 - 1 \\\ {\text{D}}{\text{. }}\sqrt 2 \\\$

Answer 1

Hint: - Use trigonometric identity ${\text{cos}}\left( {90^\circ - \theta } \right) = \sin \theta$

As given in the question let’s first solve the given expression: -
$\Rightarrow \sin \theta + \cos \theta = \sqrt 2 \cos \left( {90^\circ - \theta } \right)$
Therefore, above expression will become
$\Rightarrow \sin \theta + \cos \theta = \sqrt 2 \sin \theta {\text{ }}$ as we know
${\text{cos}}\left( {90^\circ - \theta } \right) = \sin \theta$
$\Rightarrow {\text{so,cos}}\theta {\text{ = }}\left( {\sqrt 2 - 1} \right)\sin \theta \\\ \Rightarrow \cot \theta = \sqrt 2 - {\text{1}}{\text{.}} \\\$

Note: - Whenever this kind of question appears always first simplify the equation as much as
possible. Remember in this type of question basic knowledge of trigonometric identities is
must. Remember $\cos \theta$ always remain positive in the fourth quadrant.