Here is the completed proof of [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex] with the correct expressions and reasons filled in:
According to the Pythagorean Identity, it is known that [tex]\(a^2 + b^2 = 1\)[/tex].
By the definition of sine, [tex]\(\sin \theta = \frac{b}{c}\)[/tex].
By the definition of cosine, [tex]\(\cos \theta = \frac{a}{c}\)[/tex].
Therefore, by combining like terms, [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex].
So the completed proof is:
According to the Pythagorean Identity it is known that [tex]\(a^2 + b^2 = 1\)[/tex].
By the definition of sine, [tex]\(\sin \theta = \frac{b}{c}\)[/tex].
By the definition of cosine, [tex]\(\cos \theta = \frac{a}{c}\)[/tex].
Therefore, by combining like terms, [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex].
