Question

Explain the method of using fundamental trigonometric identities to determine the simplified form of the expression?

Answer 1

The fundamental trigonometric identities are used to establish other relationships among trigonometric functions. To verify an identity we show that one side of the identity can be simplified so that is identical to the other side. Each side is manipulated independently of the other side of the equation. Usually it is best to start with the more complicated side of the identity.

Complete step by step solution:
The fundamental trigonometric identities are the basic identities.

They are the reciprocal identities, the Pythagorean identities and the quotient identities.
Reciprocal identities are given below:

$$\sin a = \dfrac{1}{{\cos eca}} \\\ \cos a = \dfrac{1}{{\sec a}} \\\ \tan a = \dfrac{1}{{\cot a}} \\\ \cos esca = \dfrac{1}{{\sin a}} \\\ \sec a = \dfrac{1}{{\cos a}} \\\ \cot a = \dfrac{1}{{\tan a}} \\\$$

Pythagorean identities are given below:

$${\sin ^2}a + {\cos ^2}a = 1 \\\ 1 + {\tan ^2}a = {\sec ^2}a \\\ 1 + {\cot ^2}a = \cos e{c^2}a \\\$$

Quotient identities are given below:

$$\tan a = \dfrac{{\sin a}}{{\cos a}} \\\ \cot a = \dfrac{{\cos a}}{{\sin a}} \\\$$

When it comes down to simplifying with these identities, we must use a combination of these identities to reduce a much more complex expression to its simplest form.

Note: Work with each side of the equation should be done independently of the other side. From the more complicated side it should be started and should be transformed in a step by step fashion until it looks exactly like the other side. The identity must be analyzed and should look for opportunities to apply the fundamental identities.

Rewriting the more complicated side of the equation in terms of $\sin es$and $\cos ines$is often helpful.

If sums or differences of fractions appear on one side, the least common denominators must be used and then fractions should be combined. Each side should be manipulated independently of the other side of the equation.