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.