Simplify sin x plus sin x cotx equals cscx?

Answer 1

First convert everything to sines and cosines:

sin x + sin x cos x / sin x = 1 / sin x

sin x + cos x = 1 / sin x

Multiplying by sin x:

sin2x + sin x cos x = 1

Using the identity sin2 + cos2x = 1:

sin2x + sin x cos x = sin2x + cos2x

sin x cos x = cos2x

Dividing by cos x:

sin x = cos x

The solution is therefore x = pi / 4 radians, or x = 5 pi / 4 radians.

The division by cos x assumed that cos x was not equal to zero; this possibility must be explored in the original equation. When cos x = 0, sin x = 1 or -1, and the angle x = pi/2 or -pi/2. It seems both of these are solutions, too.

First convert everything to sines and cosines:

sin x + sin x cos x / sin x = 1 / sin x

sin x + cos x = 1 / sin x

Multiplying by sin x:

sin2x + sin x cos x = 1

Using the identity sin2 + cos2x = 1:

sin2x + sin x cos x = sin2x + cos2x

sin x cos x = cos2x

Dividing by cos x:

sin x = cos x

The solution is therefore x = pi / 4 radians, or x = 5 pi / 4 radians.

The division by cos x assumed that cos x was not equal to zero; this possibility must be explored in the original equation. When cos x = 0, sin x = 1 or -1, and the angle x = pi/2 or -pi/2. It seems both of these are solutions, too.

First convert everything to sines and cosines:

sin x + sin x cos x / sin x = 1 / sin x

sin x + cos x = 1 / sin x

Multiplying by sin x:

sin2x + sin x cos x = 1

Using the identity sin2 + cos2x = 1:

sin2x + sin x cos x = sin2x + cos2x

sin x cos x = cos2x

Dividing by cos x:

sin x = cos x

The solution is therefore x = pi / 4 radians, or x = 5 pi / 4 radians.

The division by cos x assumed that cos x was not equal to zero; this possibility must be explored in the original equation. When cos x = 0, sin x = 1 or -1, and the angle x = pi/2 or -pi/2. It seems both of these are solutions, too.

First convert everything to sines and cosines:

sin x + sin x cos x / sin x = 1 / sin x

sin x + cos x = 1 / sin x

Multiplying by sin x:

sin2x + sin x cos x = 1

Using the identity sin2 + cos2x = 1:

sin2x + sin x cos x = sin2x + cos2x

sin x cos x = cos2x

Dividing by cos x:

sin x = cos x

The solution is therefore x = pi / 4 radians, or x = 5 pi / 4 radians.

The division by cos x assumed that cos x was not equal to zero; this possibility must be explored in the original equation. When cos x = 0, sin x = 1 or -1, and the angle x = pi/2 or -pi/2. It seems both of these are solutions, too.

Answer 2

First convert everything to sines and cosines:

sin x + sin x cos x / sin x = 1 / sin x

sin x + cos x = 1 / sin x

Multiplying by sin x:

sin2x + sin x cos x = 1

Using the identity sin2 + cos2x = 1:

sin2x + sin x cos x = sin2x + cos2x

sin x cos x = cos2x

Dividing by cos x:

sin x = cos x

The solution is therefore x = pi / 4 radians, or x = 5 pi / 4 radians.

The division by cos x assumed that cos x was not equal to zero; this possibility must be explored in the original equation. When cos x = 0, sin x = 1 or -1, and the angle x = pi/2 or -pi/2. It seems both of these are solutions, too.