Proof that Sine Squared Plus Cosine Squared Equals 1
Formulas
$$ \sin^2\theta+\cos^2\theta=1 $$
Proof
1-Addition Formula for Cosine
Using the Addition Theorem for Cosine, we can understand it very easily.
$$ \cos(\theta_{1}-\theta_2)=\cos\theta_{1}\cos\theta_2 + \sin\theta_{1}\sin\theta_2 $$
Here, if we substitute $\theta$ instead of $\theta_{1}$, $\theta_2$
$$\cos(\theta-\theta)=\cos^2\theta + \sin^2\theta$$
$$\implies \cos(\theta-\theta)=\cos 0=1$$
$$\implies \sin^2\theta+\cos^2\theta=1$$
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2-Pythagorean Theorem
There is a unit circle with a radius of 1. Let’s look at the triangle formed by the unit circle’s radius, the perpendicular dropped to the $x$ axis from the point of tangency on the circle, and the $x$ axis itself. The length of the base is $\cos\theta$, the length of the height is $\sin\theta$, and the length of the hypotenuse is $1$. Therefore, by the Pythagorean Theorem
$$ \sin^2\theta+\cos^2\theta=1 $$
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