Review – Verifying Trigonometric Identities

To verify a trigonometric identity, you need to use existing trigonometric identities to gradually modify one side of the equation and eventually make it look exactly the same as the other side.

Some tips for verifying identities are the following:

• Try to express everything in sin's and cos's.
• Only work with one side of the equation.

Here is an example of a verifying problem; I will list out the steps as I solve it:

cos(–θ) / [1+sin(–θ)] = secθ + tanθ

Use the Even and Odd identities to simplify the terms containing –θ:

cosθ / (1 – sinθ) =

Multiply the numerator and denominator with the conjugate of the denominator:

[cosθ × (1 + sinθ)] / [(1 – sinθ) × (1 + sinθ)] =

Distribute:

(cosθ + cosθsinθ) / (1 – sin²θ) =

Use the Pythagorean identity sin²θ + cos²θ = 1 to change the denominator to cos²θ:

(cosθ + cosθsinθ) / cos²θ =

Split this fraction into two fractions:

(cosθ / cos²θ) + (cosθsinθ / cos²θ) =

Cancel out the cos's:

(1 / cosθ) + (sinθ / cosθ) =

Use the Reciprocal identity 1 / cosθ = secθ and the Quotient identity sinθ / cosθ = tanθ to rewrite the fractions:

secθ + tanθ = secθ + tanθ

Since the LHS (left-hand side) is now equal to the RHS (right-hand side), the verification is done!