Trigonometry: Power reducing identities along Proof and Examples

In mathematics, power reducing identities are used for the reduction of the powers of trigonometry functions. These identities are widely used in trigonometry. The working of these identities is very simple as they have used some formulas to reduce the powers of the trigonometry functions.

The reduction of the power like squared or cubic trigonometry functions are to be reduced in smaller powers

In this article, we will discuss all the power-reducing identities like definition, working, proof, and a lot of examples.

What are power-reducing identities?

To write the squared or cubic trigonometry functions in smaller powers. The power-reducing identities are very helpful in this regard. It is also helpful for simplifying trigonometric expressions. It is also helpful for the reductions of the powers in the differentiation of the trigonometry terms in calculus.

In this reduction, the formulas are used like double angle or half-angle for the simplification of the trigonometric terms. In these formulas, the functions of trigonometry are raised in the powers.

By using the formulas of the trigonometry any function can be reduce to lower power. These formulas are written as

Squared identities.

• sin²θ = [1 – cos (2θ)] / 2
• cos²θ = [1 + cos (2θ)] / 2
• tan²θ = [1 – cos (2θ)] / [1 + cos (2θ)]
• cosec²θ = 2 / [1 – cos (2θ)]
• sec²θ = 2 / [1 + cos (2θ)]
• cot²θ = [1 + cos (2θ)] / [1 – cos (2θ)]

Cubic identities

• sin³θ = [3sin(θ) – sin (3θ)] / 4
• cos²θ = [3cos(θ) + cos (3θ)] / 4
• tan²θ = [3sin(θ) – sin (3θ)] / [3cos(θ) + cos (3θ)]
• cosec²θ = 4 / [3sin(θ) – sin (3θ)]
• sec²θ = 4 / [3cos(θ) + cos (3θ)]
• cot²θ = [3cos(θ) + cos (3θ)] / [3sin(θ) – sin (3θ)]

These formulas are calculated by using the double and half angle identities.

1. The double angle identities are

Sin2x = 2sin(x)cos(x)

Cos2x= 2cos²x -1

tan2x = (2tanx) / (1 – tan²x) 

• And the half angle identities are

Sin (θ/2) = ± √(1-cosθ)/2

Cos (θ/2) = ± √(1+cosθ)/2

Tan (θ/2) = ± Sin(θ) / 1 +Cos(θ)

Proofs of the power reducing identities

Proof 1:

Step 1:By subtractingthe first two identities of the power reducing.

Cos²(x) – sin²(x) = [1 + cos (2θ)] / 2 – [1 – cos (2θ)] / 2

Cos²(x) – sin²(x) = ½ (1 + cos(2θ) – 1 + cos(2θ))

Cos²(x) – sin²(x) = 1/2 (2cos(2θ))

Cos²(x) – sin²(x) = 2cos(2θ)

Step 2:Now use the above term as.

[1 + cos (2θ)] / 2 = [1 + cos²(x) – sin²(x)] / 2

Step 3:Now put 1 = sin²(x) + cos²(x).

[1 + cos (2θ)] / 2 = [sin²(x) + cos²(x) + cos²(x) – sin²(x)] / 2

[1 + cos (2θ)] / 2 = [cos²(x) + cos²(x)] / 2

[1 + cos (2θ)] / 2 = [2cos²(x)] / 2

[1 + cos (2θ)] / 2 = cos²(x)

Hence, we proved the identity by using two identities.

Proof 2:

Step 1:Divide the first two identities.

sin²(θ) / Cos²(θ) = [1 – cos (2θ)] / 2/ [1 + cos (2θ)] / 2

Step 2:We know that the sin²(θ) / Cos²(θ) = tan²(θ).

Tan²(θ) = [1 – cos (2θ)] / 2/ [1 + cos (2θ)] / 2

Tan²(θ) = 2/2[1 – cos (2θ)] / [1 + cos (2θ)]

Tan²(θ) = [1 – cos (2θ)] / [1 + cos (2θ)]

Hence, by using two identities we proved the third identity.

How to reduce the powers by using power-reducing identities?

By using the formulas of these identities, we can calculate the reducing power of the trigonometric expressions. The power reducing identities calculator is also used for the calculation of these identities. You can easily find the reducing powers of each trigonometric function by using this tool.

Example 1: For Squared terms

If the angle is 60-degree, find all the squared identities of the trigonometry?

Solution

Step 1:Identify the angle.

Theta = θ = 60-degree

Step 2:Now Take the formulas.

• sin²θ = [1 – cos (2θ)] / 2
• cos²θ = [1 + cos (2θ)] / 2
• tan²θ = [1 – cos (2θ)] / [1 + cos (2θ)]

Step 3:Now put theta in each term one by one.

• sin²θ = [1 – cos (2θ)] / 2

sin²(60) = [1 – cos (2(60))] / 2

sin²(60) =[1 – cos (120)] / 2

sin²(60) =[1 – (-0.5)] / 2

sin²(60) =[1 + 0.5] / 2

sin²(60) =[1.5] / 2

sin²(60) = 0.75

• cos²θ = [1 + cos (2θ)] / 2

cos²(60) = [1 + cos (2(60))] / 2

cos²(60) = [1 + cos (120)] / 2

cos²(60) = [1 + (-0.5)] / 2

cos²(60) = [1 – 0.5] / 2

cos²(60) = [0.5] / 2

cos²(60) = 0.25

• tan²θ = [1 – cos (2θ)] / [1 + cos (2θ)]

tan²(60) = [1 – cos (2(60))] / [1 + cos (2(60))]

tan²(60) = [1 – cos (120)] / [1 + cos (120)]

tan²(60) = [1 – (-0.5)] / [1 + (-0.5)]

tan²(60) = [1 + 0.5] / [1 – 0.5]

tan²(60) = [1.5] / [0.5]

tan²(60) = 3

Step 4:Write the results collectively.

sin²(60) = 0.75

cos²(60) = 0.25

tan²(60) = 3

By taking the reciprocal of each result, you can easily find the other three trigonometric functions.

Example 2: For cubic terms

If the angle is 45-degree, find all the cubic identities of the trigonometry?

Solution

Step 1:Identify the angle.

Theta = θ = 45-degree

Step 2:Now Take the formulas.

• sin³θ = [3sin(θ) – sin (3θ)] / 4
• cos²θ = [3cos(θ) + cos (3θ)] / 4
• tan²θ = [3sin(θ) – sin (3θ)] / [3cos(θ) + cos (3θ)]

Step 3:Now put theta in each term one by one.

• sin³θ = [3sin(θ) – sin (3θ)] / 4

sin³(45) = [3sin (45) – sin (3(45))] / 4

sin³(45)= [3sin (45) – sin (135)] / 4

sin³(45)= [3(0.707) – (0.7071)] / 4

sin³(45) = [2.1213 – 0.7071] / 4

sin³(45) =1.4142 / 4

sin³(45)= 0.3535

Step 4:Similarly, you can easily identify the other terms.

Cos³(45)= 0.3535

Tan³(45)= 1

Summary

Power reducing identities are very essential in trigonometry for reducing the powers of the trigonometry functions for making the calculations easier. By following the above examples, you can solve any advanced problems related to this easily.