You’ve just learned how to verify trigonometric identities by changing one side of an equation to match the form of the other side. You can also verify trigonometric identities by changing both sides of an equation to the same form.
Click each tab to see an example. Keep in mind that for each of these problems, one way to verify the equation is shown, but there may be multiple ways to verify the equation.
Verify that the trigonometric equation \( \frac{\sec x + 1}{\tan x} = \frac{\sin x}{1 - \cos x} \) is an identity.
Cross multiply since the equation is written as a proportion. |
The equation now becomes \( (\sec x + 1)\left( 1 - \cos x \right) = \tan x \sin x \). |
Multiply the left side of the equation. |
The equation now becomes \( \sec x - \cos x \sec x + 1 - \cos x = \tan x \sin x \). |
Apply the reciprocal identity with secant. |
Use the secant identity, \( \sec\theta = \frac{1}{\cos\theta} \) \( \frac{1}{\cos x} = \sec x \) Substitute in \( \frac{1}{\cos x} \) for \( \sec x \). \( \frac{1}{\cos x} - \cos x \frac{1}{\cos x} + 1 - \cos x = \tan x \sin x \) |
Simplify the equation. |
The equation now becomes \( \frac{1}{\cos x} - 1 + 1 - \cos x = \tan x \sin x \). Then by combining like terms, the equation becomes \( \frac{1}{\cos x} - \cos x = \tan x \sin x \). |
Apply the quotient identity with tangent. |
Use the tangent quotient identity, \( \tan\theta = \frac{\sin\theta}{\cos\theta} \) \( \tan x = \frac{\sin x}{\cos x} \) Substitute in \( \frac{\sin x}{\cos x} \) for \( \tan x \). \( \frac{1}{\cos x} - \cos x = \frac{\sin x}{\cos x} \sin x \) |
Multiply both sides of the equation by \( \cos x \) and simplify the equation. |
The equation becomes \( 1 - \cos^{2} x = \sin^{2} x \) |
Apply the Pythagorean identity with sines and cosines. |
Use the Pythagorean identity with sines and cosines, \( \sin^{2}\theta + \cos^{2}\theta = 1 \) \( \sin^{2} x + \cos^{2} x = 1 \) \( \sin^{2} x = 1 - \cos^{2} x \) Substitute in \( \sin^{2} x \) for \( 1 - \cos^{2} x \). \( \sin^{2} x = \sin^{2} x \) |
The equation is proven as an identity since the left side and right side of the equation are equal.
Verify that the trigonometric equation \( \frac{\cos x}{1 + \sin x} = \sec x - \tan x \) is an identity.
Apply the reciprocal identity with secant. |
Use the secant identity, \( \sec\theta = \frac{1}{\cos\theta} \) \( \frac{1}{\cos x} = \sec x \) Substitute in \( \frac{1}{\cos x} \) for \( \sec x \). \( \frac{\cos x}{1 + \sin x} = \frac{1}{\cos x} - \tan x \) |
Apply the quotient identity for tangent. |
Use the tangent quotient identity, \( \tan\theta = \frac{\sin\theta}{\cos\theta} \) \( \tan x = \frac{\sin x}{\cos x} \) Substitute in \( \frac{\sin x}{\cos x} \) for \( \tan x \). \( \frac{\cos x}{1 + \sin x} = \frac{1}{\cos x} - \frac{\sin x}{\cos x} \) |
Write the left side of the equation as one rational expression. |
\( \frac{\cos x}{1 + \sin x} = \frac{1 - \sin x}{\cos x} \) |
Cross multiply since the equation is written as a proportion. |
The equation now becomes \( (1 + \sin x)\left( 1 - \sin x \right) = \cos^{2} x \). |
Multiply the left side of the equation. |
\( 1 - \sin^{2} x = \cos^{2} x \) |
Apply the Pythagorean identity with sines and cosines. |
Use the Pythagorean identity with sines and cosines, \( \sin^{2}\theta + \cos^{2}\theta = 1 \) \( \sin^{2} x + \cos^{2} x = 1 \) \( \cos^{2} x = 1 - \sin^{2} x \) Substitute in \( \cos^{2} x \) for \( 1 - \sin^{2} x \). \( \cos^{2} x = \cos^{2} x \) |
The equation is proven as an identity since the left side and right side of the equation are equal.
Level Up!
Complete the activity below to practice verifying trigonometric identities by changing both sides of an equation to the same form. 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 be multiple ways to verify the equation.
Verify that the trigonometric equation \( \frac{\sin x \csc x}{\cot x} = \tan x \) is an identity.
Cross multiply the equation since it is written as a proportion. |
\( \sin x \csc x = \cot x \tan x \). |
Apply the quotient identity for the tangent and cotangent. |
\( \sin x \csc x = \left( \frac{\cos x}{\sin x} \right)\left( \frac{\sin x}{\cos x} \right) \). |
Simplify the right side of the equation. |
\( \sin x \csc x = 1 \) |
Apply the reciprocal identity for cosecant. |
\( \sin x \left( \frac{1}{\sin x} \right) = 1 \) |
Simplify the left side of the equation. |
\( 1 = 1 \) |
The equation is proven as an identity since the left side and right side of the equation are equal.
Verify that the trigonometric equation \( \frac{\cos x}{1 + \sin x} + \frac{1 + \sin x}{\cos x} = 2\sec x \) is an identity.
Apply the reciprocal identity for secant. |
\( \frac{\cos x}{1 + \sin x} + \frac{1 + \sin x}{\cos x} = \frac{2}{\cos x} \) |
Multiply all terms of the equation by \( \cos x \). |
\( \frac{\cos^{2} x}{1 + \sin x} + 1 + \sin x = 2 \) |
Subtract 1 from both sides of the equation. |
\( \frac{\cos^{2} x}{1 + \sin x} + \sin x = 1 \) |
Multiply all terms of the equation by \( 1 + \sin x \). |
\( \cos^{2} x + \sin x(1 + \sin x) = 1 + \sin x \) |
Perform the multiplication. |
\( \cos^{2} x + \sin x + \sin^{2} x = 1 + \sin x \) |
Apply the Pythagorean identity with sine and cosine. |
\( 1 + \sin x = 1 + \sin x \) |
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.
