1 + cot^2 x
To simplify the expression 1 + cot^2(x), we need to use the trigonometric identity that relates cotangent to cosine:
cot^2(x) = 1 / tan^2(x) = 1 / (sin^2(x) / cos^2(x)) = cos^2(x) / sin^2(x)
Now we can substitute this expression into the original expression:
1 + cot^2(x) = 1 + cos^2(x) / sin^2(x)
Using a common denominator, we can combine the fractions:
1 + cos^2(x) / sin^2(x) = (sin^2(x) + cos^2(x)) / sin^2(x)
Since sin^2(x) + cos^2(x) = 1 (which is a fundamental trigonometric identity), the expression simplifies to:
1 + cot^2(x) = 1 / sin^2(x)
Therefore, the simplified expression is 1 / sin^2(x)
To simplify the expression 1 + cot^2(x), we need to use the trigonometric identity that relates cotangent to cosine:
cot^2(x) = 1 / tan^2(x) = 1 / (sin^2(x) / cos^2(x)) = cos^2(x) / sin^2(x)
Now we can substitute this expression into the original expression:
1 + cot^2(x) = 1 + cos^2(x) / sin^2(x)
Using a common denominator, we can combine the fractions:
1 + cos^2(x) / sin^2(x) = (sin^2(x) + cos^2(x)) / sin^2(x)
Since sin^2(x) + cos^2(x) = 1 (which is a fundamental trigonometric identity), the expression simplifies to:
1 + cot^2(x) = 1 / sin^2(x)
Therefore, the simplified expression is 1 / sin^2(x).
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