Math Doubts

Cos squared formula

Expansion form

$\cos^2{\theta} \,=\, 1-\sin^2{\theta}$

Simplified form

$1-\sin^2{\theta} \,=\, \cos^2{\theta}$

How to use

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

  1. The square of cos function is expanded as subtraction of sin squared function from one.
  2. The subtraction of sin squared function from one is simplified as square of cos function.

Proof

The cosine squared formula is derived from the Pythagorean identity of sin and cos functions.

If angle of a right triangle is theta, then the sum of squares of sin and cos functions is equal to one.

$\sin^2{\theta}+\cos^2{\theta} \,=\, 1$

$\therefore \,\,\,\,\,\, \cos^2{\theta} \,=\, 1-\sin^2{\theta}$

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

Alternative form

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

For example, if angle of right angled triangle is denoted by $x$, then the cosine squared formula is written as $\cos^2{x} \,=\, 1-\sin^2{x}$

Remember, the angle of right triangle can be represented by any symbol, the cosine squared formula should be written in terms of the corresponding symbol.



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