# Proving the identity of trigonometric

The eight basic identities are used to prove other identities. To prove an identity means to show that the two sides of the equation are always equivalent. It is generally more efficient to work with the more complicated side of the identity and show, by using the basic identities and algebraic principles, that the two sides are the same.

Tips for Proving an Identity

To prove an identity, use one or more of the following tips:

• Work with the more complicated side of the equation.
• Use basic identities to rewrite unlike functions in terms of the same function.
• Remove parentheses.
• Find common denominators to add fractions.
• Simplify complex fractions and reduce fractions to lowest terms.

Example 1:

Prove that $\sec \theta \sin \theta =\tan \theta$ is an identity.

Solution:

Write the left side of the equation in terms of $\sin \theta$ and $\cos \theta$.

Proof begins with what is known and proceeds to what is to be proved. Although we have written the proof in Example 1 by starting with what is to be proved and ending with what is obviously true, the proof of this identity really begins with the obviously true statement:

$\tan \theta=\tan \theta$ ; therefore $\sec \theta \sin \theta =\tan \theta$

Example 2:

Prove that $\sin \theta(\csc \theta -\sin \theta)=\cos^2\theta$.

Solution:

Use the distributive property to simplify the left side.

The Pythagorean identity $\cos^2\theta +\sin^2\theta =1$ can be rewritten as $\cos^2\theta =1-\sin^2\theta$. The second to last line of the proof is often omitted and the left side, $1-\sin^2\theta$, replaced by $\cos^2\theta$.

Also Read : The basic identities of trigonometric

Example 3:

Prove the identity $1-\sin \theta =\frac{\cos^2\theta}{1+\sin \theta}$

Solution:

For this identity, it appears that we need to multiply both sides of the equation by ($1+\sin \theta$) to clear the denominator. However, in proving an identity we perform only operations that change the form but not the value of that side of the equation.

Here we will work with the right side because it is more complicated and multiply by $\frac{1-\sin \theta}{1-\sin \theta}$, a fraction equal to 1.

×