What is a simplified form of the expression cos^3(theta) + cos(theta)/csc^2(theta)
very confused lol
cos T ( ONE) = cos T
cos^2 angle + sin^2 angle = ONE
in right triangle
c^2 = a^2 + b^2
cos angle at A = b/c
sin angle at A = a/c
so cos^2 a = b^2/c^2
and sin^2 a = a^2/c^2
sum = a^2/c^2 + b^2/c^2 = (a^2+b^2)/c^2
but c^2 = a^2+b^2
SO
sum = (a^2+b^2) /(a^2+b^2) = ONE !!!!
Well, let me tell you a joke to make simplifying this expression more fun:
Why did the cosine go to the party alone?
Because sine and tangent were taken, and they couldn't find a secant!
Now, let's simplify the expression:
First, let's rewrite cos^3(theta) as (cos(theta))^3. Then, let's convert cos(theta) and csc(theta) to their respective forms:
cos(theta) = 1/sin(theta)
csc(theta) = 1/sin(theta)
Now, we can rewrite the expression as:
(cos(theta))^3 + cos(theta) / (1/sin^2(theta))
Simplifying further, we get:
(cos(theta))^3 + cos(theta) * sin^2(theta)
And that's a simplified form of the expression!
To simplify the expression cos^3(theta) + cos(theta)/csc^2(theta), let's start by rewriting the trigonometric functions using their reciprocal identities.
Reciprocal identities:
- csc(theta) = 1/sin(theta)
- sec(theta) = 1/cos(theta)
- cot(theta) = 1/tan(theta)
Now, let's substitute these reciprocal identities into the expression:
cos^3(theta) + cos(theta)/csc^2(theta)
= cos^3(theta) + cos(theta)/(1/sin^2(theta))
To simplify further, let's convert the division into multiplication by flipping the fraction:
cos^3(theta) + cos(theta) * sin^2(theta)
Next, let's use the identity sin^2(theta) = 1 - cos^2(theta) to simplify the expression:
cos^3(theta) + cos(theta) * (1 - cos^2(theta))
= cos^3(theta) + cos(theta) - cos^3(theta)
By combining like terms, the two terms with cos^3(theta) cancel each other out, leaving us with the simplified form:
cos(theta)
cos^3 T + cos T/csc^2 T
csc = 1/sin
so
cos^3 T + cos T sin^2 T
oh my look
cos T ( cos^2 T + sin^2 T) but we all know what cos^2+sin^2 is !