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Express \[{\mathbf{cot}}{\text{ }}{\mathbf{85}}^\circ + {\text{ }}{\mathbf{cos}}{\text{ }}{\mathbf{75}}^\circ \] in terms of trigonometric ratios of angles between 0° and 45°.

Last updated date: 24th Jun 2024
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Hint: Trigonometric identities are useful whenever trigonometric functions are involved in an expression or an equation. Geometrically, these identities involve certain functions of one or more angles. The trigonometric identities hold true only for the right-angle triangle.
Trigonometric identities are formulas that involve trigonometric ratios of all the angles. These identities are true for all values of the variables.
Here, we use the complementary angles identities. These formulas are used to shift the angles. They are also called as co-function identities.
\cos \left( {90^\circ - \theta } \right) = \sin \theta \\
\cot \left( {90^\circ - \theta } \right) = \tan \theta \\
\end{gathered} $
All trigonometric identities are cyclic in nature. They repeat themselves after this periodicity constant. This periodicity constant is different for different trigonometric identities.

Complete step by step solution: We know that
  \cos \left( {90^\circ - \theta } \right) = \sin \theta \\
  \cot \left( {90^\circ - \theta } \right) = \tan \theta \\
\end{gathered} $
Therefore, we can write $\cot 85^\circ + \cos 75^\circ $ as
  \cot 85^\circ + \cos 75^\circ = {\text{cot}}\left( {90^\circ - 5^\circ } \right) + \cos \left( {90^\circ - 15^\circ } \right) \\
   = \tan 5^\circ + \sin 15^\circ \\
\end{gathered} $
Hence, the required value is $\tan 5^\circ + \sin 15^\circ $

Note: Trigonometry is a branch of mathematics which deals with the measurement of sides and angles of a triangle and the problems based on them. Trigonometry helps us to find angles and distances, and is used a lot in science, engineering, and many more. There are many trigonometry formulas and trigonometric identities, which are used to solve complex equations in geometry.
The ratios of sides of a right-angled triangle with respect to any of its acute angles are known as the trigonometric ratios of that particular angle. They are defined by parameters namely hypotenuse, base and perpendicular. Trigonometric identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.