How is the tangent function expressed in terms of complementary angles?

The relationship involving the tangent function and complementary angles is best expressed through the identity that relates the tangent of an angle to the cotangent of its complement. Specifically, the identity states that the tangent of an angle ( x ) is the cotangent of the angle ( (π/2 - x) ). This can be mathematically represented as:

[

\tan\left(\frac{\pi}{2} - x\right) = \cot(x)

]

This identity arises from the definitions of the tangent and cotangent functions in terms of sine and cosine. The tangent of an angle is defined as the ratio of the sine of that angle to the cosine of that angle:

[

\tan(x) = \frac{\sin(x)}{\cos(x)}

]

For the complement, ( \tan\left(\frac{\pi}{2} - x\right) ) can be reformulated using the co-function identities of sine and cosine, where:

[

\sin\left(\frac{\pi}{2} - x\right) = \cos(x) \quad \text{and} \quad \cos\left(\frac{\pi}{2} - x\right) = \sin(x)

\