The power-reducing formulas in trigonometry are used to express powers of sine, cosine, tangent and cotangent in terms of their first powers. These formulas are derived from the double-angle identities. Here are the power-reducing formulas:

1) For cosine:

2) For sine:

3) For cotangent:

4) For tangent:

These formulas are particularly useful when integrating or simplifying expressions involving higher powers of sine, cosine, tangent and cotangent. By reducing the power of the trigonometric functions, they often make calculations more manageable.

How the power reducing formulas in trigonometric functions are obtained?

The power-reducing formulas for sine and cosine are derived from the double-angle identities for cosine. Let's derive them step by step:

1) Deriving the power reducing formula for cosine:

Step 1: Writing the Double-Angle Identity for Cosine:

From Trigonometry, it is known that the double-angle identity for cosine is

Step 2: Using the Pythagorean Identity:

we write

Step 3: Substituting the last expression into the Double-Angle Identity for cosine yields:

This gives:

For example, given that cos 90⁰ = 0, then

Indeed, since cos 45⁰ = √2/2, when raising it to the second power yields:

Since there is the same result obtained, the formula is confirmed.

2) Deriving the power reducing formula for sine:

Step 1: Using again the Pythagorean Identity to express sin² (θ) in terms of cos² (θ), yields:

Step 2: Substitute the power-reducing formula for cos2(θ) into the above equation:

For example, given that cos 60⁰ = 1/2, we can write:

Indeed, since sin (30⁰) = 1/2, then

Since there is the same result obtained, the formula is confirmed.

3) Deriving the power reducing formula for cotangent:

Given that

we substitute in the above equation the formulas found above for the power reducing of cosine and sine. Hence,

Simplifying 2 from both numerator and denominator yields:

For example, since cos 120⁰ = -1/2, we have for cot² (60⁰):

Indeed, since cos 60⁰ = 1/2 and sin 60⁰ = √3/2, we can write:

Since there is the same result obtained, the formula is confirmed.

4) Deriving the power reducing formula for tangent:

Given that

we substitute in the above equation the same formulas as above for the power reducing of cosine and sine. We have,

Simplifying 2 from both numerator and denominator yields:

For example, since cos 120⁰ = -1/2, we have for tan² (60⁰):

Indeed, since cos 60⁰ = 1/2 and sin 60⁰ = √3/2, we can write:

Since there is the same result obtained, the formula is confirmed.

In summary, by using the double-angle identity for cosine and the Pythagorean identity, we can derive the power-reducing formulas for cosine, sine cotangent and tangent. These formulas allow us to express higher powers of sine and cosine in terms of their first powers, which can be useful for simplifying expressions and solving integrals.

Question

What is the power reducing formula in trigonometry?

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