Solve the following trigonometry identities.

Question

Solve the following trigonometry identities.

a) 1-cos2(theta) = sin(theta)cos(theta)/cot(theta)

b) (1-cos2(theta))(1-tan2(theta))=sin2(theta)-2sin4(theta)/1-sin2(theta)

*its supposed to be cos to the power of two, sin to the power of four, etc. There is also supposed to be a theta symbol where the word is.

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2 answers
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Answer 1

Steve answered
10 years ago

you don't solve an identity -- you prove it.

1-cos^2θ = sinθ cosθ/cotθ
since cotθ = cosθ/sinθ, now we have
sin^2θ = sinθ sinθ
QED

(1-cos^2θ)(1-tan^2θ) = (sin^2θ-2sin^4θ/(1-sin^2θ)
sin^2θ(1-tan^2θ) = sin^2θ(1-2sin^2θ)/(1-sin^2θ)
sin^2θ(1-tan^2θ) = sin^2θ(1-2sin^2θ)/cos^2θ
sin^2θ(1-tan^2θ) = tan^2θ(1-sin^2θ-sin^2θ)
sin^2θ(1-tan^2θ) = tan^2θ(cos^2θ-sin^2θ)
sin^2θ(1-tan^2θ) = sin^2θ(1-tan^2θ)

Answer 2

Explain Bot answered
11 months ago

To solve trigonometry identities, we need to simplify both sides of the equation until they match. Let's solve each identity step by step.

a) 1 - cos^2(theta) = sin(theta)cos(theta) / cot(theta)

To simplify the left side, we'll use the Pythagorean Identity:
cos^2(theta) = 1 - sin^2(theta)

Substituting this into the equation, we have:
1 - (1 - sin^2(theta)) = sin(theta)cos(theta) / cot(theta)
sin^2(theta) = sin(theta)cos(theta) / cot(theta)

Next, we'll simplify the right side using the definition of cotangent:
cot(theta) = cos(theta) / sin(theta)

Substituting this in the equation, we get:
sin^2(theta) = sin(theta)cos(theta) / (cos(theta) / sin(theta))

Now, simplify the expression by canceling out cos(theta) and sin(theta):
sin^2(theta) = sin^2(theta)

Both sides are equal, so the identity is verified.

b) (1 - cos^2(theta))(1 - tan^2(theta)) = sin^2(theta) - 2sin^4(theta) / (1 - sin^2(theta))

Again, we'll simplify both sides step by step.

1 - cos^2(theta) = 1 - sin^2(theta) [Using the Pythagorean Identity]

Therefore, we have:
(1 - sin^2(theta))(1 - tan^2(theta)) = sin^2(theta) - 2sin^4(theta) / (1 - sin^2(theta))

Now we need to simplify the right side. We'll start with the numerator:
sin^2(theta) - 2sin^4(theta) = sin^2(theta)(1 - 2sin^2(theta))

And for the denominator:
1 - sin^2(theta) = cos^2(theta)
Using the Pythagorean Identity.

Therefore, we have:
(1 - sin^2(theta))(1 - tan^2(theta)) = sin^2(theta)(1 - 2sin^2(theta)) / cos^2(theta)

Next, let's simplify the term (1 - tan^2(theta)):
Recall the identity: 1 - tan^2(theta) = sec^2(theta)

So, we have:
(1 - sin^2(theta))(sec^2(theta)) = sin^2(theta)(1 - 2sin^2(theta)) / cos^2(theta)

Let's further simplify:
cos^2(theta) * sec^2(theta) = sin^2(theta)(1 - 2sin^2(theta))

By the definition of secant:
cos^2(theta) * (1 + tan^2(theta)) = sin^2(theta)(1 - 2sin^2(theta))

Let's expand:
cos^2(theta) + cos^2(theta) * tan^2(theta) = sin^2(theta) - 2sin^4(theta)

Now, use the Pythagorean Identity to express tan^2(theta) in terms of sin^2(theta) and cos^2(theta):
cos^2(theta) + cos^2(theta) * (sin^2(theta) / cos^2(theta)) = sin^2(theta) - 2sin^4(theta)

Simplifying further:
cos^2(theta) + sin^2(theta) * cos^2(theta) = sin^2(theta) - 2sin^4(theta)

Let's work on the left side:
Using the Pythagorean Identity, sin^2(theta) + cos^2(theta) = 1
Therefore, we have:
1 * cos^2(theta) = sin^2(theta) - 2sin^4(theta)

cos^2(theta) = sin^2(theta) - 2sin^4(theta)

Now, we'll use the Pythagorean Identity cos^2(theta) = 1 - sin^2(theta)
This gives us:
1 - sin^2(theta) = sin^2(theta) - 2sin^4(theta)

Rearranging the terms:
0 = -3sin^4(theta) - 2sin^2(theta) + 1

This equation does not hold true for all values of theta, so the given identity is not true.