Let's Break It Down
Verifying that a trigonometric equation is an identity means making the two sides of the equation identical to each other in order to prove that it is true.
You can verify trigonometric identities by changing one or both sides of the equation until they’re the same. Here, you will learn how to verify trigonometric identities by changing one side of an equation to match the form of the other side. On the next page, you will learn how to verify trigonometric identities by changing both sides of an equation to the same form.
Either way, the overall process is to make one side look exactly like the other using a combination of trigonometric identities (e.g., reciprocal, quotient, Pythagorean) and algebra techniques (e.g., factoring, rewriting expression with a common denominator). If these approaches do not work, rewrite all trigonometric ratios in terms of sines and cosines.
Click each tab to see an example. Keep in mind that for each of these problems, only one way to verify the identity shown. There may be multiple ways to verify the identity.
Verify that the trigonometric equation \( \sec^{2} x - \sec^{2} x \sin^{2} x = 1 \) is an identity.
Factor \( \sec^{2} x \) on the left side of the equation. |
The equation now becomes \( \sec^{2} x \left( 1 - \sin^{2} x \right) = 1 \). |
Apply the Pythagorean identity with sines and cosines. |
Use the Pythagorean identity with sines and cosines, \( \sin^{2}\theta + \cos^{2}\theta = 1 \) \( \sin^{2} x + \cos^{2} x = 1 \) \( \sin^{2} x = 1 - \cos^{2} x \) Substitute in \( \cos^{2} x \) for \( 1 - \sin^{2} x \). \( \sec^{2} x \left( \cos^{2} x \right) = 1 \) |
Apply the reciprocal identity for secant. |
Use the secant identity, \( \sec\theta = \frac{1}{\cos\theta} \) \( \sec^{2} x = \frac{1}{\cos^{2} x} \) Substitute in \( \frac{1}{\cos^{2} x} \) for \( \sec^{2} x \). \( \frac{1}{\cos^{2} x}\left( \cos^{2} x \right) = 1 \) |
Simplify the left side of the equation. |
Simplify the left side of the equation. \( \frac{1}{\cos^{2} x}\left( \cos^{2} x \right) = 1 \) \( 1 = 1 \) |
The equation is proven as an identity since the left side and right side of the equation are equal.
Verify that the trigonometric equation \( \frac{1 + \cos x}{\sin x} = \csc x + \cot x \) is an identity.
Rewrite the left side of the equation as two fractions. |
The equation now becomes \( \frac{1}{\sin x} + \frac{\cos x}{\sin x} = \csc x + \cot x \). |
Apply the reciprocal identity with cosecant. |
Use the cosecant identity, \( \csc\theta = \frac{1}{\sin\theta} \) \( \frac{1}{\sin x} = \csc x \) Substitute in \( \csc x \) for \( \frac{1}{\sin x} \). \( \csc x + \frac{\cos x}{\sin x} = \csc x + \cot x \) |
Apply the quotient identity for cotangent. |
Use the cotangent quotient identity, \( \cot\theta = \frac{\cos\theta}{\sin\theta} \) \( \cot x = \frac{\cos x}{\sin x} \) Substitute in \( \cot x \) for \( \frac{\cos x}{\sin x} \). \( \csc x + \cot x = \csc x + \cot x \) |
The equation is proven as an identity since the left side and right side of the equation are equal.
Verify that the trigonometric equation \( \sin x(\tan x + \cot x) = \sec x \) is an identity.
Apply the distributive property on the left side of the equation. |
The equation now becomes \( \sin x \tan x + \sin x \cot x = \sec x \). |
Rewrite the left side of the equation with sines and cosines. |
Use the quotient identity for tangent and cotangent, \( \tan\theta = \frac{\sin\theta}{\cos\theta} \) and \( \cot\theta = \frac{\cos\theta}{\sin\theta} \) \( \tan x = \frac{\sin x}{\cos x} \) and \( \cot x = \frac{\cos x}{\sin x} \) \( \sin x \left( \frac{\sin x}{\cos x} \right) + \sin x \left( \frac{\cos x}{\sin x} \right) = \sec x \) |
Simplify the left side of the equation by multiplication. |
\( \frac{\sin^{2} x}{\cos x} + \cos x = \sec x \) |
Rewrite the left side of the equation with a common denominator of \( \cos x \) for each term. |
\( \frac{\sin^{2} x}{\cos x} + \cos x \left( \frac{\cos x}{\cos x} \right) = \sec x \) \( \frac{\sin^{2} x}{\cos x} + \frac{\cos^{2} x}{\cos x} = \sec x \) \( \frac{\sin^{2}{x + \cos^{2} x}}{\cos x} = \sec x \) |
Apply the Pythagorean identity with sines and cosines. |
Use the Pythagorean identity with sines and cosines, \( \sin^{2}\theta + \cos^{2}\theta = 1 \) \( \sin^{2} x + \cos^{2} x = 1 \) Substitute in 1 for \( \sin^{2} x + \cos^{2} x. \) \( \frac{1}{\cos x} = \sec x \) |
Apply the reciprocal identity with cosecant. |
Use the cosecant identity, \( \sec\theta = \frac{1}{\cos\theta} \) \( \frac{1}{\cos x} = \sec x \) Substitute in \( \sec x \) for \( \frac{1}{\cos x} \). \( \sec x = \sec x \) |
The equation is proven as an identity since the left side and right side of the equation are equal.
Show Your Skills!
Complete the activity below to practice verifying trigonometric identities by changing one side of an equation to match the form of the other side. Verify the trigonometric equation on each tab then check your answer. For each of these problems, one way to verify the equation is shown, but there may by multiple ways to verify the equation.
Verify that the trigonometric equation \( \csc^{2} x \sin^{2} x - \sin^{2} x = \cos^{2} x \) is an identity.
Factor \( \sin^{2} x \) on the left side of the equation. |
\( \sin^{2} x(\csc^{2} x - 1) = \cos^{2} x \). |
Apply the Pythagorean identity with cosecant and secant. |
\( \sin^{2} x(\cot^{2} x) = \cos^{2} x \). |
Apply the quotient identity for cotangent. |
\( \sin^{2} x \left( \frac{\cos^{2} x}{\sin^{2} x} \right) = \cos^{2} x \) |
Simplify the left side of the equation by multiplication. |
\( \cos^{2} x = \cos^{2} x \) |
The equation is proven as an identity since the left side and right side of the equation are equal.
Verify that the trigonometric equation \( \frac{1 + \sin x}{\cos x} = \sec x + \tan x \) is an identity.
Write the left side of the equation as two fractions. |
\( \frac{1}{\cos x} + \frac{\sin x}{\cos x} = \sec x + \tan x \) |
Apply the reciprocal identity for secant. |
\( \sec x + \frac{\sin x}{\cos x} = \sec x + \tan x \) |
Apply the quotient identity for tangent. |
\( \sec x + \tan x = \sec x + \tan x\) |
The equation is proven as an identity since the left side and right side of the equation are equal.
Verify that the trigonometric equation: \( \frac{\tan x + \sin x}{1 + \cos x} = \tan x \) is an identity.
Apply the quotient identity for tangent. |
\( \frac{\frac{\sin x}{\cos x} + \sin x}{1 + \cos x} = \tan x \) |
Rewrite the numerator with a common denominator. |
\( \frac{\frac{\sin x}{\cos x} + \sin x \left( \frac{\cos x}{\cos x} \right)}{1 + \cos x} = \tan x \). |
Write the numerator as one fraction. |
\( \frac{\frac{\sin x + \sin x \cos x}{\cos x}}{1 + \cos x} = \tan x \) |
Factor the numerator. |
\( \frac{\frac{\sin{x(1} + \cos x)}{\cos x}}{1 + \cos x} = \tan x \) |
Multiply by the reciprocal of denominator and simplify. |
\( \frac{\sin x}{\cos x} = \tan x \) |
Apply the quotient identity for tangent. |
\( \tan x = \tan x \) |
The equation is proven as an identity since the left side and right side of the equation are equal.
