Verifying trigonometric identities through examples with detailed solutions are presented.
No single method works for all identities. However following certain steps might help. To verify an identity, you may start by transforming the more complicated side into the other, using basic trigonometric identities. You may also transform the two sides into one same expression.
More questions with solutions are included.
Example 1
Verify the identity \[ \cos x \cdot \tan x = \sin x \]
Solution to Example 1:
Transform the left side using the identity \( \tan x = \dfrac{\sin x}{\cos x} \).
\[ \cos x \cdot \tan x = \cos x \cdot \dfrac{\sin x}{\cos x} \]
Simplify to obtain.
\[ = \sin x \]
The left side was transformed so that it is equal to the right side.
Example 2
Verify the identity \[ \cot x \cdot \sec x \cdot \sin x = 1 \]
Solution to Example 2
Use the identities \( \cot x = \dfrac{\cos x}{\sin x} \) and \( sec x = \dfrac{1}{\cos x} \) to rewrite the left side as
\[ \cot x \cdot \sec x \cdot \sin x = \dfrac{\cos x}{\sin x} \cdot \dfrac{1}{\cos x} \cdot \sin x \]
Simplify the right side to obtain.
\[ = 1 \]
The left side is transformed so that it is equal to the right side.
Example 3
Verify the identity
\[ \dfrac{\cot x - \tan x }{ \sin x \cdot \cos } = \csc^2 x - \sec^2 x \]
Solution to Example 3
Use the identities \( \cot x = \dfrac{\cos x}{\sin x} \) and \( \tan x = \dfrac{\sin x}{\cos x} \) to transform the left side as follows.
\[ \dfrac{\cot x - \tan x }{ \sin x \cdot \cos } = \dfrac {\dfrac{\cos x}{\sin x} - \dfrac{\sin x}{\cos x}}{\sin x \cdot \cos x} \]
Rewrite the numerator of the main fraction on the right as the difference of two fractions with a common denominator.
\[ = \dfrac {\dfrac{\cos x \cos x }{\cos x \sin x} - \dfrac{\sin x \sin x}{\sin x \cos x}}{\sin x \cdot \cos x} \]
Group the fractions in the numerator on the right side.
\[ = \dfrac {\dfrac{\cos^2 x - \sin^2 x }{\cos x \cdot \sin x}}{\sin x \cdot \cos x} \]
Divide the above and rewrite as.
\[ = \dfrac {\cos^2 x - \sin^2 x }{\sin^2 x \cdot \cos^2 x} \]
Rewrite as a difference of two fractions.
\[ = \dfrac {\cos^2 x}{\sin^2 x \cdot \cos^2 x} - \dfrac {\sin^2 x }{\sin^2 x \cdot \cos^2 x} \]
Simplify each fraction.
\[ = \dfrac {1}{\sin^2 x} - \dfrac {1}{\cos^2 x} \]
We now transform the above side using the identities \( \quad \csc^2 x = \dfrac{1}{\sin^2 x} \) and \( \quad \sec^2 x = \dfrac{1}{\cos^2 x } \).
\[ = \csc^2 x - \sec^2 x \]
The left side has been transformed so that it is equalt to the right side and hence the identity is verified.