Math and Arithmetic
Trigonometry

# How do you simplify csc theta cot theta?

###### Wiki User

There are 6 basic trig functions.

sin(x) = 1/csc(x)

cos(x) = 1/sec(x)

tan(x) = sin(x)/cos(x) or 1/cot(x)

csc(x) = 1/sin(x)

sec(x) = 1/cos(x)

cot(x) = cos(x)/sin(x) or 1/tan(x)

---- In your problem csc(x)*cot(x) we can simplify csc(x).

csc(x) = 1/sin(x)

Similarly, cot(x) = cos(x)/sin(x).

csc(x)*cot(x) = (1/sin[x])*(cos[x]/sin[x])

= cos(x)/sin2(x) = cos(x) * 1/sin2(x)

Either of the above answers should work.

In general, try converting your trig functions into sine and cosine to make things simpler.

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## Related Questions

For a start, try converting everything to sines and cosines.

cosec(q)*cot(q)*cos(q) = 1/sin(q)*cot(q)*cos(q) = cot2(q)

Since sin(theta) = 1/cosec(theta) the first two terms simply camcel out and you are left with 1 divided by tan(theta), which is cot(theta).

By converting cosecants and secants to the equivalent sine and cosine functions. For example, csc theta is the same as 1 / sin thetha.

That depends on the value of the angle, theta. csc is short for "cosecans", and is the reciprocal of the sine. That is, csc theta = 1 / sin theta.

It's easiest to show all of the work (explanations/identities), and x represents theta. cosxcotx + sinx = cscx cosx times cosx/sinx + sinx = csc x (Quotient Identity) cosx2 /sinx + sinx = csc x (multiplied) 1-sinx2/sinx + sinx = csc x (Pythagorean Identity) 1/sinx - sinx2/sinx + sinx = csc x (seperate fraction) 1/sinx -sinx + sinx = csc x (canceled) 1/sinx = csc x (cancelled) csc x =csc x (Reciprocal Identity)

'csc' = 1/sin'tan' = sin/cosSo it must follow that(cos) (csc) / (tan) = (cos) (1/sin)/(sin/cos) = (cos) (1/sin) (cos/sin) = (cos/sin)2

There can be no significant simplicfication if some of the angles are theta and others are x, so assume that all angles are x. [csc(x) - cot(x)]*[cos(x) + 1] =[1/sin(x) - cos(x)/sin(x)]*[cos(x) + 1] =1/sin(x)*[1 - cos(x)]*[cos(x) + 1] =1/sin(x)*[1 - cos2(x)] =1/sin(x)*[sin2(x)] = sin(x)

The derivative of csc(x) is -cot(x)csc(x).

Cotan(theta) is the reciprocal of the tan(theta). So, cot(theta) = 1/2.

sec(x)*cot(x) = (1/cos(x))*(cos(x)/sin(x)) = (1/sin(x)) = csc(x)

If tan(theta) = x then sin(theta) = x/(sqrt(x2 + 1) so that csc(theta) = [(sqrt(x2 + 1)]/x = sqrt(1 + 1/x2)

To simplify such expressions, it helps to express all trigonometric functions in terms of sines and cosines. That is, convert tan, cot, sec or csc to their equivalent in terms of sin and cos.

&int;cscxcotx*dx&int;csc(u)cot(u)*du= -csc(u)+C, where C is the constant of integrationbecause d/dx(csc(u))=-[csc(u)cot(u)],so d/dx(-csc(u))=csc(u)cot(u).&int;cscxcotx*dxLet:u=xdu/dx=1du=dx&int;cscucotu*du= -csc(u)+CPlug in x for u.&int;cscxcotx*dx= -csc(x)+C

The integral for csc(u)dx is -ln|csc(u) + cot(u)| + C.

With all due respect, you don't really want to know howto solve it.You just want the solution.csc(&Icirc;&tilde;) = 1/sin(&Icirc;&tilde;)tan(&Icirc;&tilde;) = sin(&Icirc;&tilde;)/cos(&Icirc;&tilde;)csc(&Icirc;&tilde;) x tan(&Icirc;&tilde;) = 1/sin(&Icirc;&tilde;) x sin(&Icirc;&tilde;)/cos(&Icirc;&tilde;) = 1/cos(&Icirc;&tilde;) = sec(&Icirc;&tilde;)

By converting everything to sines and cosines. Since tan x = sin x / cos x, in the cotangent, which is the reciprocal of the tangent: cot x = cos x / sin x. You can replace any other variable (like thetha) for the angle.

tan2(theta) + 5*tan(theta) = 0 => tan(theta)*[tan(theta) + 5] = 0=> tan(theta) = 0 or tan(theta) = -5If tan(theta) = 0 then tan(theta) + cot(theta) is not defined.If tan(theta) = -5 then tan(theta) + cot(theta) = -5 - 1/5 = -5.2

From math class, some trigonometric identities: cot x = 1/tan x csc x = 1/sin x sec x = 1/cos x There are no built-in cot or csc formulas, so use the above. Remember that these give errors when tan x, sin x, or cos x are equal to 0.

Depending on your calculator, you should have an arcsin function, which appears as sin^-1. It's usually a 2nd function of the sin key. If you don't have this function, there are many free calculators you can download... just google scientific calculator downloads.Anyway, this inverse function will give you theta when you plug in the value of sin theta. Here's the algebra written out:sin(theta)=-0.0138arcsin(sin(theta))=arcsin(-0.0138)theta=.......The inverse function applied to both sides of the equation "cancels out" the sin function and yields the value of the angle that was originally plugged into the function, in this case theta. You can use this principle to solve for theta for any of the other trig functions:arccos(cos(theta))=thetaarctan(tan(theta))=thetaand so on, but calculators usually only have these three inverse functions, so if you encounter a problem using sec, csc, or cot, you need to rewrite it as cos, sin, or tan.sec=1/coscsc=1/sincot=1/tan