Question

How do you use the fundamental trigonometric identities to determine the simplified form of the expression?

Answer 1

Trigonometric identities:

The fundamental trigonometric identities are the basic identities:

The reciprocal identities are defined as follows:

$$\begin{gathered} \sin A=\frac{1}{\cos ecA},\cos A=\frac{1}{secA},\tan A=\frac{1}{cotA} \\ \csc A=\frac{1}{\sin A},\sec A=\frac{1}{\cos A},\cot A=\frac{1}{\tan A} \end{gathered}$$
The quotient identities are defined as follows:

$\tan A=\frac{\sin A}{\cos A},\cot A=\frac{\cos A}{\sin A}$

The Pythagorean identities:

$$\begin{gathered} {\sin }^{2}A+{\cos }^{2}A=1 \\ 1+{\tan }^{2}A={\sec }^{2}A \\ 1+{\cot }^{2}A={\mathrm{cosec}}^{2}A \end{gathered}$$

The trigonometric identities help to reduce the complex identities into a simpler form.

Consider an example:

$\frac{\tan \beta +cot\beta }{\cos e{c}^2\beta }$

Simplify the the expression using trigonometric identities:

$\begin{array}{rcl}\frac{\tan \beta +cot\beta }{\cos e{c}^2\beta }& =& \frac{\frac{\sin \beta }{\cos \beta }+\frac{\cos \beta }{\sin \beta }}{\cos e{c}^2\beta }\left[\because \tan A=\frac{\sin A}{\cos A},\cot A=\frac{\cos A}{\sin A}\right]\\ & =& \frac{\frac{\sin \beta }{\cos \beta }+\frac{\cos \beta }{\sin \beta }}{\frac{1}{{\sin }^2\beta }}\left[\because \cos ecA=\frac{1}{\sin A}\right]\\ & =& \frac{\frac{{\sin }^2\beta +{\cos }^2\beta }{\cos \beta .\sin \beta }}{\frac{1}{{\sin }^2\beta }}\\ & =& \frac{\frac{1}{\cos \beta .\sin \beta }}{\frac{1}{{\sin }^2\beta }}\left[\because {\sin }^{2}A+{\cos }^{2}B=1\right]\\ & =& \frac{1}{\cos \beta .\sin \beta }\times {\sin }^2\beta \\ & =& \frac{\sin \beta }{\cos \beta }\left[\because \frac{\sin \beta }{\cos \beta }=\tan \beta \right]\\ & =& \tan \beta \end{array}$

Thus $\begin{array}{rcl}\frac{\tan \beta +cot\beta }{\cos e{c}^2\beta }& =& \tan \beta \end{array}$ after simplification of the expression using trigonometric identities