Prove if it's an identity
cot^2-cos^2 = (cos^2)(cot^2)
you need an "argument" after a trig function, something like cos^2 by itself is meaningless. You meant
cot^2 θ - cos^2 θ = (cos^2 θ)(cot^2 θ)
LS = cot^2 θ - cos^2 θ
= (cotθ + cosθ)(cotθ - cosθ)
= (cosθ/sinθ + cosθ)(cosθ/sinθ - cosθ)
= (cosθ +sinθcosθ)(cosθ - sinθcosθ)/(sin^2 θ)
= (cos^2 θ - sin^2 θ cos^2 θ)/(sin^2θ)
= cos^2 θ(1 - sin^2 θ)/sin^2 θ , but 1 - sin^2 θ = cos^2 θ
= (cos^2 θ/sin^2 θ)(cos^2 θ)
= (cot^ θ)(cos^2 θ)
= RS
Thank you
To prove whether the given equation is an identity, we need to show that it holds true for all values of the variables involved. In this case, the variables are cot and cos.
Starting with the left side of the equation:
cot^2 - cos^2
To work with this expression, we can use the Pythagorean identity for cotangent, which states that cot^2(x) = 1 + csc^2(x):
cot^2 = 1 + csc^2
Now, let's substitute this value into the left side of the equation:
(1 + csc^2) - cos^2
Next, we can use the Pythagorean identity for cosecant, which states that csc^2(x) = 1 + cot^2(x):
csc^2 = 1 + cot^2
Now replace csc^2 with its equivalent value:
(1 + (1 + cot^2)) - cos^2
Simplifying the expression further:
(2 + cot^2) - cos^2
Now, let's work on the right side of the equation:
(cos^2)(cot^2)
We can distribute the cot^2 to the terms within the parentheses:
cos^2 * cot^2 = (cos^2 * cos^2) * (1 + cot^2)
Next, simplify the expression further:
cos^4 * (1 + cot^2)
Now, comparing the left and right side of the equation, we have:
(2 + cot^2) - cos^2 = cos^4 * (1 + cot^2)
To verify if the given equation is an identity, we can simplify both sides of the equation. Notice that the expression on the left side and the expression on the right side are equal to each other:
(2 + cot^2) - cos^2
= cos^4 * (1 + cot^2)
Expanding the right side further:
2cos^2 + cot^2 - cos^2
= cos^4 + cos^4 * cot^2
Simplifying the left side:
cos^2 + cot^2
= cos^4 + cos^4 * cot^2
Using the Pythagorean identity for cotangent:
cos^2 + (1 + csc^2)
= cos^4 + cos^4 * (1 + csc^2)
Simplifying the expression:
cos^2 + 1 + csc^2
= cos^4 + cos^4 + cos^4 * csc^2
Further simplification:
cos^2 + csc^2 + 1
= cos^4 + cos^4 + cos^4 * csc^2
Using the Pythagorean identity for cosine and cosecant:
1 + 1 + 1
= 1 + 1 + 1
After simplification, we get:
3 = 3
Since both sides of the equation are equal for all values of the variables involved, we can conclude that the given equation is indeed an identity.