Math Doubts

Product identity of Tangent and Cotangent functions

Formula

$\tan{\theta}\cot{\theta} \,=\, 1$

Proof

The tangent and cotangent functions are reciprocal function mathematically. Therefore, the product of them equals to one and the product relation between tan and cot functions can also be proved in trigonometric mathematics.

$\Delta QPR$ is a right triangle and the angle of this triangle is theta ($\theta$).

right triangle

Write Tan function in ratio form

Express tangent function ($\tan{\theta}$) in its ratio form firstly.

$\tan{\theta} \,=\, \dfrac{QR}{PR}$

Write Cot function in ratio form

After that, write cotangent function ($\cot{\theta}$) in its ratio form.

$\cot{\theta} \,=\, \dfrac{PR}{QR}$

Evaluate Product of Tangent and Cotangent

Lastly, multiply both tan and cot functions to obtain product of them.

$\tan{\theta} \times \cot{\theta}$ $\,=\,$ $\dfrac{QR}{PR} \times \dfrac{PR}{QR}$

$\implies$ $\tan{\theta} \times \cot{\theta}$ $\,=\,$ $\dfrac{QR \times PR}{PR \times QR}$

$\implies$ $\tan{\theta} \times \cot{\theta}$ $\,=\,$ $\require{cancel} \dfrac{\cancel{QR} \times \cancel{PR}}{\cancel{PR} \times \cancel{QR}}$

$\,\,\, \therefore \,\,\,\,\,\,$ $\tan{\theta}.\cot{\theta} \,=\, 1$

Therefore, it is proved mathematically that the product of tan and cot functions at an angle is equal one, and this property is used in trigonometry as a formula.

Note

One important thing is the angle of a right triangle can be denoted by any symbol but the product of tangent of angle and cotangent of angle always equals to one.

For example, if $x$ represents angle of right triangle, then

$\tan{x}.\cot{x} \,=\, 1$

Similarly, if $A$ is used to denote angle of right triangle, then

$\tan{A}.\cot{A} \,=\, 1$

The tangent and cotangent functions in the product trigonometric identity can be written in terms of any angle but the product of them equals to one mathematically.

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