Verify that (sec theta/sin theta)(cot theta/csc theta) = csc theta is an identity.
Thanks
calling theta t
(1/cos t)/ sin t * (cos t/sin t) *sin t
(1/(cos t sin t) * cos t
1/sin t
= csc t
To verify that the expression (sec(theta) / sin(theta)) * (cot(theta) / csc(theta)) is an identity, we need to simplify it and show that it is equal to csc(theta).
Let's start by simplifying the expression step by step:
1. Recall that the reciprocal of a trigonometric function can be found by interchanging sine and cosine, cosine and secant, and tangent and cotangent.
Using this property, we can rewrite the expression as:
(sec(theta) / sin(theta)) * (cot(theta) / csc(theta))
= (1 / cos(theta)) * (cos(theta) / (1 / sin(theta)))
2. Next, we can simplify the double reciprocal by canceling out the common factors:
= (1 / cos(theta)) * (cos(theta) * sin(theta))
= 1 * sin(theta)
3. Finally, we know that sin(theta) divided by 1 is equal to csc(theta). Therefore, we can conclude that:
(sec(theta) / sin(theta)) * (cot(theta) / csc(theta))
= csc(theta)
Thus, we have successfully shown that the given expression is an identity, as the simplified form is equal to csc(theta).