Math Doubts

Tan squared formula

Expansion form

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

Simplified form

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

How to use

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

  1. The square of tan function is expanded as subtraction of one from secant squared function.
  2. The subtraction of one from secant squared function is simplified as square of tan function.

Proof

The tangent squared formula is derived from the Pythagorean identity of secant and tan functions.

If angle of a right triangle is theta, then the subtraction of squares of tan function from sec function is equal to one.

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

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

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

Therefore, it is proved that tan squared theta is equal to the subtraction of one from sec squared theta.

Alternative form

The tangent squared identity is often expressed in terms of different angles.

For example, if $x$ is used to denote angle of right triangle, then the tan squared formula is written as $\tan^2{x} \,=\, \sec^2{x}-1$

Keep it mind that the angle of right triangle can be represented by any symbol, the tan squared formula must be written in terms of the respective symbol.



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