Simplifying the Expression 1 + cot^2 x | A Trigonometric Identity

1 + cot^2 x

To simplify the expression 1 + cot^2 x, we need to use trigonometric identities

To simplify the expression 1 + cot^2 x, we need to use trigonometric identities.

First, let’s recall the definition of the cotangent (cot) function. The cotangent of an angle is the ratio of the adjacent side to the opposite side in a right triangle. Mathematically, cot x = adjacent side / opposite side = cos x / sin x.

Now, let’s take a look at the Pythagorean identity. The Pythagorean identity states that sin^2 x + cos^2 x = 1. By rearranging this equation, we can solve for cos^2 x by subtracting sin^2 x from both sides:

cos^2 x = 1 – sin^2 x

Substituting this into our cot equation:

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

Now, let’s square both sides of this equation to eliminate the fraction:

(cot x)^2 = [(1 – sin^2 x) / sin x]^2

Expanding the numerator:

(cot x)^2 = [(1 – sin^2 x)^2 / sin^2 x]

Simplifying the numerator:

(cot x)^2 = [(1 – 2sin^2 x + sin^4 x) / sin^2 x]

Let’s simplify further:

(cot x)^2 = (1 – 2sin^2 x + sin^4 x) / sin^2 x

Now, let’s add 1 to both sides of this equation:

1 + (cot x)^2 = 1 + (1 – 2sin^2 x + sin^4 x) / sin^2 x

Distribute 1 to each term in the numerator:

1 + (cot x)^2 = (sin^2 x + 1 – 2sin^2 x + sin^4 x) / sin^2 x

Combine like terms in the numerator:

1 + (cot x)^2 = (1 – sin^2 x + sin^4 x) / sin^2 x

Notice that the numerator is the same as the numerator in our cot equation!

Therefore, we can conclude that:

1 + (cot x)^2 = cot^2 x

So, 1 + cot^2 x simplifies to cot^2 x.

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