Question

True or False:${\sin }^2\left(\theta \right)=1-{\cos }^2\left(\theta \right)$

Short answer

The given identity is true

Step-by-step solution

Given information

We are given a identity${\sin }^2\left(\theta \right)=1-{\cos }^2\left(\theta \right)$

Explanation for true statement

We know a trigonometric identity

${\sin }^2\left(\theta \right)+{\cos }^2\left(\theta \right)=1$

on rearranging the identity we get,

${\sin }^2\left(\theta \right)=1-{\cos }^2\left(\theta \right)$

Hence the given identity is true

Explanation for false statement

Consider that the given identity is false

Hence,

${\sin }^2\left(\theta \right)\ne 1-{\cos }^2\left(\theta \right)$

And therefore

${\sin }^2\left(\theta \right)+{\cos }^2\left(\theta \right)\ne 1$

Which is false.

Conclusion

The given statement is true statement