express in terms of sine and cosine.

a) tan theta/ cot theta

i know the answer is sin^2 theta / cos ^2 theta but how do you do it?

ctn Theta= 1/TanTheta

tanTheta/ctnTheta= tan^2Theta= (sinT/cosT)^2

To express tan theta / cot theta in terms of sine and cosine, we need to rewrite the tangent (tan theta) and the cotangent (cot theta) in terms of sine and cosine.

The tangent of an angle theta is equal to the sine of theta divided by the cosine of theta: tan theta = sin theta / cos theta.

The cotangent of an angle theta is equal to the cosine of theta divided by the sine of theta: cot theta = cos theta / sin theta.

Now, let's substitute these expressions into tan theta / cot theta:
tan theta / cot theta = (sin theta / cos theta) / (cos theta / sin theta).

To divide fractions, we can multiply the first fraction by the reciprocal of the second fraction:
= (sin theta / cos theta) * (sin theta / cos theta).

Multiplying the numerators and denominators:
= (sin theta * sin theta) / (cos theta * cos theta).

Using the trigonometric identity sin^2 theta + cos^2 theta = 1, we can rewrite the expression further:
= (sin^2 theta) / (cos^2 theta).

So, tan theta / cot theta is equal to sin^2 theta / cos^2 theta.

To express the expression tan(theta)/cot(theta) in terms of sine and cosine, we can start by recalling the definitions of tangent and cotangent:

tan(theta) = sin(theta) / cos(theta)
cot(theta) = cos(theta) / sin(theta)

Let's substitute these definitions into the given expression:

tan(theta) / cot(theta) = (sin(theta) / cos(theta)) / (cos(theta) / sin(theta))

Now, to divide fractions, we can multiply the numerator by the reciprocal of the denominator:

tan(theta) / cot(theta) = (sin(theta) / cos(theta)) * (sin(theta) / cos(theta))

Multiplying these fractions together:

tan(theta) / cot(theta) = sin(theta) * sin(theta) / (cos(theta) * cos(theta))

Using the trigonometric identity sin^2(theta) = sin(theta) * sin(theta) and cos^2(theta) = cos(theta) * cos(theta), we can simplify further:

tan(theta) / cot(theta) = sin^2(theta) / cos^2(theta)

So, the expression tan(theta) / cot(theta) is equivalent to sin^2(theta) / cos^2(theta).