Math Doubts

Sec squared formula

Expansion form

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

Simplified form

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

How to use

The secant squared identity is used as a formula in two cases in trigonometry.

  1. The square of secant function is expanded as sum of one and tangent squared function.
  2. The sum of one and tan squared function is simplified as secant squared function.

Proof

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

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

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

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

Therefore, it is proved that secant squared theta is equal to the summation of one and tan squared theta.

Alternative form

The secant squared identity is also often written in terms of different angles.

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

Hence, the angle of right angled triangle can be denoted by any symbol, the sec squared formula must be written in terms of the corresponding symbol.



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