Math Doubts

Cot squared formula

Expansion form

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

Simplified form

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

How to use

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

  1. The square of cot function is expanded as subtraction of one from co-secant squared function.
  2. The subtraction of one from cosecant squared function is simplified as square of cot function.

Proof

The cotangent squared formula is derived from the Pythagorean identity of cosecant and cot functions.

If theta ($\theta$) is taken as angle of a right triangle, then the subtraction of squares of cot function from cosecant function is equal to one.

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

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

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

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

Alternative form

The cotangent squared identity is sometimes expressed in terms of various angles.

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

Keep one thing in your mind that the angle of right triangle can be represented by any symbol, the cot squared formula should be written in terms of the respective symbol.



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