Worked example
Simplify sin(−x)cos(−x) + tan(−x)
Answer
−sin x cos x − tan x
Steps
- sin(−x)cos(−x) + tan(−x)Original.
- = (−sin x)(cos x) + (−tan x)Sine and tangent are odd, cosine is even. · Sine is odd, Cosine is even, Tangent is odd
- = −sin x cos x − tan xCollect the signs.
- = −(1/2)(2 sin x cos x) − tan xOptional next step: prepare the double-angle form. · Double-angle identity for sine
- = −(1/2)sin 2x − tan xFinal compact form.
Identities used
sin(−θ) = −sin θ
cos(−θ) = cos θ
tan(−θ) = −tan θ
- Sine is odd — A negative angle inside sine: pull the minus sign out front. This is the standard first move when a problem says sin(−x) and the answers are written with positive x.
- Cosine is even — Drop the minus sign inside a cosine outright. Useful when simplifying cos(−x) + sin(−x), where the cosine keeps its sign and the sine does not.
- Tangent is odd — Pull the minus out of tan(−θ). Also the quickest way to see why tan is odd: sin flips, cos does not, so the ratio flips.
Check it yourself
Open the verifier with the main identity pre-filled — substituting your own numbers into a step is the fastest way to find out which line you did not follow.