Question

Express cot $85^\circ $ + cos $75^\circ $ in terms of trigonometric ratios of angles between $0^\circ $ and $45^\circ .$

Answer 1

Hint: We need to know the conversion of basic trigonometric angles that make use of the formula of complementary angles and try to solve this.

Complete step by step answer:
We have cot $85^\circ $ + cos $75^\circ $
Writing the above angles w.r.t. $90^\circ .$
$ \Rightarrow \cot (90^\circ - 5^\circ ) + \cos (90^\circ - 15^\circ )$
$\left[ {\because \cot (90 - \theta ) = \tan \theta \;\& \;\cos (90 - \theta ) = \sin \theta } \right]$
$ \Rightarrow \tan 5^\circ + \sin 15^\circ $
Now, in the above terms the angles are between $0^\circ $ and $45^\circ .$
$\therefore \cot 85^\circ + \cos 75^\circ = \tan 5^\circ + \sin 15^\circ $

Note: The trigonometric functions cot and tan are reciprocal to each other. And similarly sin and cos are reciprocal. We can use right angle triangles or quadrants for better understanding of these angle conversions. $\left[ {\cot (90 - \theta ) = \tan \theta \;\& \;\cos (90 - \theta ) = \sin \theta } \right]$
In the first quadrant all trigonometric functions are positive.