Solution:

To prove the given trigonometric identity, we need to simplify the left-hand side (LHS) and show that it is equal to the right-hand side (RHS). Let's start by simplifying the LHS step by step.

Step 1: Expand the expression on the LHS using the distributive property:

Sin theta(1 tan theta) cos theta(1 cot theta) = sin theta * cos theta * (1/tan theta) * (1/cot theta)

Step 2: Simplify the expression inside the parentheses:

(1/tan theta) = cot theta
(1/cot theta) = tan theta

Substituting these values into the expanded expression, we get:

LHS = sin theta * cos theta * cot theta * tan theta

Step 3: Use the trigonometric identity: cot theta * tan theta = 1

LHS = sin theta * cos theta * 1
LHS = sin theta * cos theta

Step 4: Use the trigonometric identity: sin theta * cos theta = 1/(cosec theta)

LHS = 1/(cosec theta)

Now, we have simplified the LHS to the expression 1/(cosec theta). To complete the proof, we need to show that this is equal to the RHS, which is cosec theta * cot theta.

Step 5: Simplify the RHS expression:

RHS = cosec theta * cot theta

Using the reciprocal identity, cot theta = 1/tan theta, we can rewrite the RHS as:

RHS = cosec theta * (1/tan theta)

Step 6: Use the trigonometric identity: cosec theta = 1/sin theta

RHS = (1/sin theta) * (1/tan theta)
RHS = 1/(sin theta * tan theta)

Step 7: Use the trigonometric identity: sin theta * tan theta = 1/cos theta

RHS = 1/(1/cos theta)
RHS = cos theta

We have now simplified the RHS to cos theta, which is equal to the simplified LHS expression. Therefore, we have proved that:

LHS = RHS
1/(cosec theta) = cos theta

Hence, the given trigonometric identity is true.