Trigonometry identities Power Reduction Calculator

The Trigonometry Identities Power Reduction Calculator computes sin2u, cos2u and tan2u for given angle


Calculator

 

using following formulas:

sin2u = 1/2 - (1/2)cos(2u))

cos2u = 1/2 + (1/2)cos(2u))

tan2u = (1 - cos(2u)) / (1 + cos(2u))

sin3u = (3/4)sinu - (1/4)sin(3u)

cos3u= (3/4)cosu + (1/4)cos(3u) and so on ..

The power reduction formulas allows to transform sin2(u) and cos2(u) into expressions that contains the first power of cosine of double argument.

These functions are in the same way as double-angle and half-angle functions. The power reduction formulas can be derived through the use of double-angle, half-angle formulas and Pythagorean Identity.

The use of a power reduction formula expresses the quantity without the exponent.

Proof the power reduction formula for sin and cosin

Proof for sin :

cos(2u) = cos2u - sin2u ............(1)

we will use the Pythagorean Identities as sin2u + cos2u = 1

so cos2u = 1 - sin2u, we can substitute the value of cos2u in equation (1) and we will get as :

cos2u = (1 - sin2u) - sin2u

cos2u = (1 - 2sin2u)

Now sustract 1 from the both sides

cos2u - 1 = (1 - 2sin2u) - 1

2sin2u = 1 - cos2u

sin2u = (1 - cos2u)/2

sin2u = 1/2(1 - cos2u)

 

Proof for cos :

cos(2u) = cos2u - sin2u ............(1)

We will use the Pythagorean Identities as sin2u + cos2u = 1

so sin2u = 1 -cos2u , we can substitute the value of sin2u in equation (1) and we will get as :

cos2u = cos2u - (1 -cos2u)

cos2u = 2cos2u - 1

Now add 1 from the both sides

cos2u + 1 = 2cos2u - 1 + 1

cos2u + 1 = 2cos2u

2cos2u = cos2u + 1

cos2u = 1/2(cos2u + 1)

Frequently Asked Questions (FAQs)

1. What is a Power Reduction Calculator?

A Power Reduction Calculator simplifies trigonometric expressions involving powers such as sin²θ, cos²θ and tan²θ using standard power reduction identities. It also provides step-by-step calculations.

2. What are the power reduction identities?

The most common power reduction formulas are:

  • sin²θ = (1 − cos2θ)/2
  • cos²θ = (1 + cos2θ)/2
  • tan²θ = (1 − cos2θ)/(1 + cos2θ)

3. Why are power reduction formulas useful?

These identities simplify trigonometric expressions and are widely used in integration, differentiation, solving trigonometric equations, Fourier analysis and engineering.

4. Does this calculator show step-by-step solutions?

Yes. The calculator displays every calculation step including the double-angle computation, cosine evaluation and final substitution into the power reduction formula.

5. Can I enter angles in degrees?

Yes. Enter any angle in degrees, and the calculator automatically evaluates the corresponding trigonometric power reduction expression.

6. What is the formula for sin²θ?

The identity is:

sin²θ = (1 − cos2θ)/2

7. What is the formula for cos²θ?

The identity is:

cos²θ = (1 + cos2θ)/2

8. What is the formula for tan²θ?

The identity is:

tan²θ = (1 − cos2θ)/(1 + cos2θ)

9. Where are power reduction identities used?

They are commonly used in calculus, trigonometry, physics, signal processing, engineering mathematics and competitive exams like JEE, GATE and university entrance tests.

10. Is this calculator free?

Yes. The Power Reduction Calculator on eGuruChela is completely free and works on desktop, tablet and mobile devices.