Simplifying the Expression 1 + cot^2x: Understanding the Trigonometric Function Cotangent

1 + cot^2x =

To simplify the expression 1 + cot^2x, we need to understand the trigonometric function cotangent (cot)

To simplify the expression 1 + cot^2x, we need to understand the trigonometric function cotangent (cot). The cotangent of an angle is defined as the ratio of the cosine of the angle to the sine of the angle. It can also be expressed as the reciprocal of the tangent function.

In terms of trigonometric identities, we know that:

cot^2x = (cos^2x)/(sin^2x)

Now, let’s substitute cot^2x in the given expression:

1 + cot^2x = 1 + (cos^2x)/(sin^2x)

To combine the terms, we need a common denominator. The common denominator for 1 and (cos^2x)/(sin^2x) is sin^2x. Let’s rewrite the expression accordingly:

(1 * sin^2x)/(sin^2x) + (cos^2x)/(sin^2x)

Now, we can add the fractions:

(sin^2x + cos^2x)/(sin^2x)

According to the Pythagorean identity sin^2x + cos^2x = 1, we can simplify further:

1/(sin^2x)

Hence, 1 + cot^2x simplifies to 1/(sin^2x), which is also equivalent to csc^2x (the reciprocal of the sine squared).

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