Chapter 7.4, Problem 1AYU

True or False ${\sin }^2\theta =1-{\cos }^2\theta$

To determine

Trigonometric identity in the form ${\text{sin}}^2\theta =1-{\text{cos}}^2\theta$ exists.

Answer to Problem 1AYU

Solution:

True. Trigonometric identity in the form ${\text{sin}}^2\theta =1-{\text{cos}}^2\theta$ exists.

Explanation of Solution

Given:

Trigonometric identity is in the form of ${\text{sin}}^2\theta =1-{\text{cos}}^2\theta$.

Calculation:

Consider a unit circle.

Equation of a unit circle $=x^2+y^2=1$   -----1

If $P\left(x,y\right)$ is a point on the unit circle that corresponds to angle $\theta$ then

$y=\text{ sin}\theta ,x=\text{ cos}\theta$

Putting these values in equation 1.

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

Subtracting ${\text{cos}}^2\theta$ on each side.

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