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.