cos(tan + cot) = csc
only simplify one side to equal csc
so far I got this far:
[((cos)(sin))/(cos)] + [((cos)(cos))/(sin)] = csc
I don't know what to do next
then on the left..
sin+cos^2/sin=sin+(1-sin^2)sin=sin+csc-sin=csc
You will need a common denominator of sinxcosx in your fraction to get
LS = cosx(sin^2x /(sinxcosx) + cos^2x/(sinxcosx))
= cosx (sin^2x + cos^2x)/(sinxcosx)
= cosx(1)/(sinxcosx)
= 1/sinx
= cscx
BTW you cannot just use the operators, there has to be some "angle" after it
e.g. to say tan = sin/cos is not acceptable and mathematical gibberish.
To simplify the left side of the equation, cos(tan + cot), we need to use trigonometric identities. In this case, we can use the identity:
tanθ = sinθ/cosθ
cotθ = cosθ/sinθ
Using these identities, we can simplify the left side of the equation step by step:
cos(tan + cot)
cos(tan) * cos(cot) - sin(tan) * sin(cot) <-- Using the identity: cos(A + B) = cos(A) * cos(B) - sin(A) * sin(B)
[cos(sin/cos)] * cos[(cos/sin)] - sin(sin/cos) * sin[(cos/sin)] <-- Substituting the identities
(cos(sin) * cos(cos)) / (cos^2) - sin(sin) * sin(cos) / (sin^2) <-- Simplifying
[cos(sin) * cos(cos)] / (cos^2) - sin(sin) * sin(cos) / (1 - cos^2) <-- Substituting the identity: 1 - sin^2 = cos^2
Now, let's simplify further:
[(cos * cos) / cos^2] - [sin * sin * cos / (1 - cos^2)]
[(cos * cos) / cos^2] - [sin * sin * cos / (1 - cos^2)]
Simplifying the expression:
1 - [sin * sin * cos / (1 - cos^2)] <-- Using the identity: cos^2 + sin^2 = 1
1 - [sin^2 * cos / (1 - cos^2)]
1 - [(1 - cos^2) * cos / (1 - cos^2)] <-- Using the identity: sin^2 = 1 - cos^2
1 - [cos - cos^3 / (1 - cos^2)]
1 - cos + cos^3 / (1 - cos^2)
So, the left side of the equation simplifies to:
1 - cos + cos^3 / (1 - cos^2)
Note that we have not yet reached the simplified form of csc. We may need to continue simplifying using additional trigonometric identities or calculus techniques.