$\tan^2{\theta} \,=\, \sec^2{\theta}-1$

The square of tan function equals to the subtraction of one from the square of secant function is called the tan squared formula. It is also called as the square of tan function identity.

The tangent functions are often involved in trigonometric expressions and equations in square form. The expressions or equations can be possibly simplified by transforming the tan squared functions into its equivalent form. Therefore, it is essential for learning the square of tan function formula to study the trigonometry further.

The tangent squared trigonometric law is actually used as a formula in two cases.

- The square of tan function is expanded as the subtraction of one from the secant squared function.
- The subtraction of one from the secant squared function is simplified as the square of tan function.

The tan squared function rule is also popularly expressed in two forms in trigonometry.

- $\tan^2{x} \,=\, \sec^2{x}-1$
- $\tan^2{A} \,=\, \sec^2{A}-1$

In this way, you can write the square of tangent function formula in terms of any angle in mathematics.

Take, the theta is an angle of a right triangle, then the tangent and secant are written as $\tan{\theta}$ and $\sec{\theta}$ respectively in trigonometry. The mathematical relationship between tan and secant functions can be written in the following mathematical form by the Pythagorean identity of tan and secant functions.

$\sec^2{\theta}-\tan^2{\theta} \,=\, 1$

$\implies$ $\sec^2{\theta}-1 \,=\, \tan^2{\theta}$

$\,\,\, \therefore \,\,\,\,\,\,$ $\tan^2{\theta} \,=\, \sec^2{\theta}-1$

Therefore, it has derived that the square of tan function is equal to the subtraction one from the square of secant function.

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