What is Power Reducing Formula?

The trigonometric power reduction identities help to simplify the expressions involving trigonometric terms with trigonometric terms of smaller powers. This becomes important in several applications such as integrating powers of trigonometric expressions in calculus.

Power reduction formulas like double-angle and half-angle formulas are used to simplify the calculations required to solve a given expression. A trigonometric function is raised to a power in these formulas. The use of this formula expresses the quantity without the exponent. In this article, we will discuss power reducing formula.

List of Important Power Reducing Formulas

1. sin² θ = (1 – cos 2θ)/2

2. cos² θ = (1 + cos 2θ)/2

3. tan² θ = (1 – cos 2θ)/(1 + cos 2θ)

4. cosec² θ = 2/(1 – cos 2θ)

5. sec² θ = 2/(1 + cos 2θ)

6. cot² θ = (1 + cos 2θ)/(1 – cos 2θ)

We derive power reduction formulas by using double-angle and half-angle formulas, and the Pythagorean Identity.

Proof:

For any angle θ, cos² θ = (1+cos 2θ)/2

sin² θ = (1-cos 2θ)/2

Subtracting above equations we get,

cos² θ-sin²θ = (2cos 2θ)/2 = cos 2θ

(1+cos 2θ)/2 = (1+cos² θ – sin² θ)/2

= (sin² θ+cos² θ+cos² θ – sin² θ)/2

= (2cos² θ)/2

= cos² θ

Half angle formulas

1. cos (θ/2) = ±√[(1+cos θ)/2]

2. sin (θ/2) = ±√[(1-cos θ)/2]

3. tan (θ/2) = ±√[(1-cos θ)/(1+cos θ)]

Solved Examples

Example 1: Express cos²θ sin²θ to the first power using power reduction identities.

Solution:

cos²θ sin²θ = (cos θ sin θ)²

= ¼ (2cos θ sin θ)²

= ¼ (sin² (2θ))

= ¼ (1-cos 4θ)/2

= ⅛ (1-cos 4θ)

Example 2: Express sin⁴x as sum of first powers of the cosines of multiple angles.

Solution: 

sin⁴x = (sin²x)²

= ((1-cos2x)/2)²

= (1-cos2x)²/4

= (1-2cos 2x+cos²2x)/4

= 1/4(1-2cos2x+(1+cos2x)/2)

= 3/8-(1/2)cos2x+(1/8)cos4x

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