Questions & Answers

Question

Answers

Answer

Verified

129k+ views

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.

$\begin{gathered}

\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.

$\begin{gathered}

\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

$\begin{gathered}

\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 $

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.