( 1 − cos 2 x ) ( 1 + cot 2 x ) = ( 1 − cos 2 x ) ( 1 + cos 2 x sin 2 x ) = ( 1 − cos 2 x ) ( sin 2 x sin 2 x + cos 2 x sin 2 x ) Find the common denominator. Recall that we first encountered these identities when defining trigonometric functions from right angles in Right Angle Trigonometry. The next set of fundamental identities is the set of reciprocal identities, which, as their name implies, relate trigonometric functions that are reciprocals of each other. The other four functions are odd, verifying the even-odd identities. To sum up, only two of the trigonometric functions, cosine and secant, are even. The cosecant function is therefore odd.įinally, the secant function is the reciprocal of the cosine function, and the secant of a negative angle is interpreted as sec ( − θ ) = 1 cos ( − θ ) = 1 cos θ = sec θ. The cosecant function is the reciprocal of the sine function, which means that the cosecant of a negative angle will be interpreted as csc ( − θ ) = 1 sin ( − θ ) = 1 − sin θ = − csc θ. Cotangent is therefore an odd function, which means that cot ( − θ ) = − cot ( θ ) cot ( − θ ) = − cot ( θ ) for all θ θ in the domain of the cotangent function. We can interpret the cotangent of a negative angle as cot ( − θ ) = cos ( − θ ) sin ( − θ ) = cos θ − sin θ = − cot θ. The cotangent identity, cot ( − θ ) = − cot θ, cot ( − θ ) = − cot θ, also follows from the sine and cosine identities. Tangent is therefore an odd function, which means that tan ( − θ ) = − tan ( θ ) tan ( − θ ) = − tan ( θ ) for all θ θ in the domain of the tangent function. We can interpret the tangent of a negative angle as tan (− θ ) = sin ( − θ ) cos (− θ ) = − sin θ cos θ = − tan θ. For example, consider the tangent identity, tan (− θ ) = −tan θ. The other even-odd identities follow from the even and odd nature of the sine and cosine functions.
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