Question

Demuestre las siguientes identidades trigonométricas.

1. secβ(1 − sen^2β) = cosβ

2. tanα ∗ cosα ∗ cscα = 1

3. (1/tanθ+cotgθ) = senθcosθ

4. (1 + cosβ)(1 − cosβ) = sen^2β

5. (cosθ/senθ+senθ) = cscθ


ayudeeeen porfavor :c

Answer 1

Respuesta:

Explicación paso a paso:

1. secβ(1 − senβ) = cosβ

secβ − (secβ)(senβ) = cosβ

secβ _ tgβ = cosβ

1/cosβ - (senβ/cosβ) = cosβ

(1-senβ)/cosβ = cosβ

(1-senβ) = cosβ²

(1-senβ) = 1 - senβ²

(1-senβ) = (1-senβ)(1+senβ)

1 = 1 + senβ

senβ = 0

2.

tanα ∗ cosα ∗ cscα = 1

(Senα/cosα) ∗ cosα ∗ 1/senα = 1

(Senαcosα)/(cosαsenα) = 1

1 = 1

3.

(1/tanθ+cotgθ) = senθcosθ

tanθ+cotgθ = (senθ/cosθ) + (cosθsenθ)

tanθ+cotgθ = (sen²θ+cos²θ)/(senθcosθ)

tanθ+cotgθ = 1/(senθcosθ)...(1)

(1/tanθ+cotgθ) = senθcosθ

1 = senθcosθ(tanθ+cotgθ)

1 = senθcosθ[1/(senθcosθ)]

1 = 1

4.

(1 + cosβ)(1 − cosβ) = sen²β

1 - cos²β = sen²β

1 = sen²β + cos²β

1 = 1.

5.

(cosθ/senθ+senθ) = cscθ

(cosθ/senθ+senθ) = 1/senθ

(cosθ+sen²θ)/senθ = 1/senθ

(cosθ+sen²θ) = 1

Cosθ = 1 - sen²θ

Cosθ = cos²θ

cosθ = 1