Write the expression as the sine, cosine, or tangent of an angle. sin 8x cos x - cos 8x sin x?

Solution:

We will use trigonometric identities to write the given expression as sine, cosine, and tangent of an angle.

sin(x - y) = sin(x) × cos(y) - cos(x) × sin(y)

Using the above identity, we can express in terms of sine of an angle.

⇒ sin (8x - x) = sin 8x × cos x - cos 8x × sin x

⇒ sin 7x = sin 8x × cos x - cos 8x × sin x

Thus, the expression can be expressed as sin 7x.

Since we know sin θ = cos [π/ 2 - θ]

The expression in terms of cosine of an angle 

⇒ sin 7x = cos [π/2 - 7x]

In terms of the tangent of an angle,

⇒ tan θ = sin θ/ cos θ

⇒ sin7x / cos[π/2 - 7x]

Write the expression as the sine, cosine, or tangent of an angle. sin 8x cos x - cos 8x sin x?

Summary:

The expression sin 8x cos x - cos 8x sin x as the sine, cosine, and tangent of an angle is sin7x, cos[π/2 - 7x] and sin7x / cos[π/2 - 7x] respectively.