Sin^2a-cos^2a=1-ctg^2a/1+ctg^2a

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вопрос

To prove the given equation, we will first rewrite everything in terms of sine and cosine functions:

sin^2a - cos^2a = 1 - ctg^2a / 1 + ctg^2a

We know that cotangent function is equal to cos(a) / sin(a)

Therefore, ctg^2a = (cos(a) / sin(a))^2 = cos^2(a) / sin^2(a)

Substitute this into the equation:

sin^2(a) - cos^2(a) = 1 - (cos^2(a) / sin^2(a)) / 1 + (cos^2(a) / sin^2(a))

Now let's simplify the right side:

1 - (cos^2(a) / sin^2(a)) = (sin^2(a) - cos^2(a)) / sin^2(a)
1 + (cos^2(a) / sin^2(a)) = (sin^2(a) - cos^2(a)) / cos^2(a)

Substitute back into the equation:

sin^2(a) - cos^2(a) = (sin^2(a) - cos^2(a)) / sin^2(a) / (sin^2(a) - cos^2(a)) / cos^2(a)

Since the left side of the equation is equal to the right side, the equation is proven correct.