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1 + tan 2 θ = 1 + ( sin θ cos θ ) 2 Rewrite left side . = ( cos θ cos θ ) 2 + ( sin θ cos θ ) 2 Write both terms with the common denominator . = cos 2 θ + sin 2 θ cos 2 θ = 1 cos 2 θ = sec 2 θ

The next set of fundamental identities is the set of even-odd identities. The even-odd identities    relate the value of a trigonometric function at a given angle to the value of the function at the opposite angle and determine whether the identity is odd or even. (See [link] ).

Even-Odd Identities
tan ( θ ) = tan θ cot ( θ ) = cot θ sin ( θ ) = sin θ csc ( θ ) = csc θ cos ( θ ) = cos θ sec ( θ ) = sec θ

Recall that an odd function    is one in which f (− x ) = − f ( x ) for all x in the domain of f . The sine function is an odd function because sin ( θ ) = sin θ . The graph of an odd function is symmetric about the origin. For example, consider corresponding inputs of π 2 and π 2 . The output of sin ( π 2 ) is opposite the output of sin ( π 2 ) . Thus,

sin ( π 2 ) = 1 and sin ( π 2 ) = sin ( π 2 ) = 1

This is shown in [link] .

Graph of y=sin(theta) from -2pi to 2pi, showing in particular that it is symmetric about the origin. Points given are (pi/2, 1) and (-pi/2, -1).
Graph of y = sin θ

Recall that an even function    is one in which

f ( x ) = f ( x )  for all  x  in the domain of  f

The graph of an even function is symmetric about the y- axis. The cosine function is an even function because cos ( θ ) = cos θ . For example, consider corresponding inputs π 4 and π 4 . The output of cos ( π 4 ) is the same as the output of cos ( π 4 ) . Thus,

cos ( π 4 ) = cos ( π 4 )                0.707

See [link] .

Graph of y=cos(theta) from -2pi to 2pi, showing in particular that it is symmetric about the y-axis. Points given are (-pi/4, .707) and (pi/4, .707).
Graph of y = cos θ

For all θ in the domain of the sine and cosine functions, respectively, we can state the following:

  • Since sin (− θ ) = sin θ , sine is an odd function.
  • Since, cos (− θ ) = cos θ , cosine is an even function.

The other even-odd identities follow from the even and odd nature of the sine and cosine functions. For example, consider the tangent identity, tan (− θ ) = −tan θ . We can interpret the tangent of a negative angle as tan (− θ ) = sin ( θ ) cos (− θ ) = sin θ cos θ = tan θ . Tangent is therefore an odd function, which means that tan ( θ ) = tan ( θ ) for all θ in the domain of the tangent function .

The cotangent identity, cot ( θ ) = cot θ , also follows from the sine and cosine identities. We can interpret the cotangent of a negative angle as cot ( θ ) = cos ( θ ) sin ( θ ) = cos θ sin θ = cot θ . Cotangent is therefore an odd function, which means that cot ( θ ) = cot ( θ ) for all θ in the domain of the cotangent function .

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 . The cosecant function is therefore odd.

Finally, 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 secant function is therefore even.

To sum up, only two of the trigonometric functions, cosine and secant, are even. The other four functions are odd, verifying the even-odd identities.

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. See [link] .

Practice Key Terms 4

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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