Math Doubts

Cosecant squared formula

Expansion form

$\csc^2{\theta} \,=\, 1+\cot^2{\theta}$

Simplified form

$1+\cot^2{\theta} \,=\, \csc^2{\theta}$

How to use

The cosecant squared identity is used as a formula in two ways in trigonometry.

  1. The square of cosecant function is expanded as sum of one and cotangent squared function.
  2. The sum of one and cot squared function is simplified as cosecant squared function.

Proof

The cosecant squared formula is originally derived from the Pythagorean identity of co-secant and cot functions.

If theta is used to denote angle of a right triangle, then the subtraction of squares of cot function from cosecant function equals to one.

$\csc^2{\theta}-\cot^2{\theta} \,=\, 1$

$\therefore \,\,\,\,\,\, \csc^2{\theta} \,=\, 1+\cot^2{\theta}$

Therefore, it is proved that cosecant squared theta equals to the summation of one and cot squared theta.

Alternative form

The cosecant squared identity is also usually written in terms of different angles.

For example, if $x$ is used to write as angle of right angled triangle, then the csc squared formula is written as $\csc^2{x} \,=\, 1+\cot^2{x}$

Remember, the angle of a right triangle can be represented by any symbol but the csc squared formula has to be written in terms of the respective symbol.



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