1=cot^2x+csc x

Find all solutions

To find all solutions to the equation 1 = cot^2x + cscx, we can rewrite the equation using trigonometric identities.

First, recall that cotangent (cot) is the reciprocal of tangent (tan). So, cot^2x can be expressed as 1/tan^2x.

Similarly, cscx is the reciprocal of sine (sin). So, we can write cscx as 1/sinx.

Now, let's substitute these values into the equation:

1 = 1/tan^2x + 1/sinx

To simplify the equation further, we need a common denominator. The common denominator is sin^2x. Multiply the first term by sin^2x/sin^2x and the second term by tan^2x/tan^2x:

1 = sin^2x / (sin^2x * tan^2x) + tan^2x / (sin^2x * tan^2x)

Now, combine the terms over the common denominator:

1 = (sin^2x + tan^2x) / (sin^2x * tan^2x)

Using the Pythagorean identity sin^2x + cos^2x = 1, we can replace sin^2x in the numerator:

1 = (1 - cos^2x + tan^2x) / (sin^2x * tan^2x)

Further simplifying the equation:

1 = (1 + tan^2x - cos^2x) / (sin^2x * tan^2x)

Rearranging the terms:

sin^2x * tan^2x = 1 + tan^2x - cos^2x

Now, let's substitute tan^2x = (sin^2x / cos^2x) into the equation:

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

Simplify further:

sin^4x / cos^2x = (1 + sin^2x - cos^2x * cos^2x) / cos^2x

Now, multiply both sides of the equation by cos^2x to eliminate the denominator:

sin^4x = 1 + sin^2x - cos^2x * cos^2x

Substituting cos^2x = 1 - sin^2x:

sin^4x = 1 + sin^2x - (1 - sin^2x) * (1 - sin^2x)

Expanding and simplifying:

sin^4x = 1 + sin^2x - (1 - 2sin^2x + sin^4x)

Notice that sin^4x appears on both sides of the equation. Let's simplify further:

0 = 2sin^2x - sin^2x

Collecting like terms:

0 = sin^2x

Since sin^2x = 0, it means that sinx = 0.

To find all solutions, we need to determine the values of x for which sinx = 0. The solutions occur when x = kπ, where k is an integer.

Therefore, the solutions to the equation 1 = cot^2x + cscx are x = kπ, where k is an integer.