Simplifying the Expression 1 + cot^2(x): A Trigonometric Approach

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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