find an expression equivalent to sec theta sin theta cot theta csc theta
secØ sinØ cotØ cscØ
= (1/cosØ)(sinØ)(cosØ/sinØ)(1/sinØ)
= 1/sinØ = cscØ
To find an expression equivalent to sec(theta)sin(theta)cot(theta)csc(theta), we can use trigonometric identities.
Step 1: Rewrite sec(theta) using the reciprocal identity: sec(theta) = 1/cos(theta).
Step 2: Rewrite cot(theta) using the reciprocal identity: cot(theta) = 1/tan(theta) = cos(theta)/sin(theta).
Step 3: Rewrite csc(theta) using the reciprocal identity: csc(theta) = 1/sin(theta).
Substituting these values into the original expression, we have:
sec(theta)sin(theta)cot(theta)csc(theta) = (1/cos(theta))sin(theta)(cos(theta)/sin(theta))(1/sin(theta)).
Step 4: Simplify the expression by canceling out common factors:
(1/cos(theta))(cos(theta)/sin(theta))(1/sin(theta)) = (1/sin(theta))(cos(theta)/cos(theta))(1/sin(theta)).
The cos(theta)/cos(theta) simplifies to 1, and we are left with:
(1/sin(theta))(1/sin(theta)) = 1/(sin(theta))^2.
Therefore, an expression equivalent to sec(theta)sin(theta)cot(theta)csc(theta) is 1/(sin(theta))^2.
To find an expression equivalent to sec(theta) sin(theta) cot(theta) csc(theta), we need to simplify it using trigonometric identities.
Let's start by rewriting the expression using reciprocal identities:
sec(theta) = 1/cos(theta)
csc(theta) = 1/sin(theta)
Now, substitute these values back into the expression:
(1/cos(theta)) * sin(theta) * cot(theta) * (1/sin(theta))
Next, simplify by canceling out the common terms:
= (1/cos(theta)) * (1/sin(theta)) * cot(theta)
We can further simplify cot(theta) by using the identity cot(theta) = cos(theta)/sin(theta):
= (1/cos(theta)) * (1/sin(theta)) * (cos(theta)/sin(theta))
Now, cancel out sin(theta) terms:
= 1 * (1/cos(theta)) * (cos(theta))/1
It simplifies to:
= cos(theta)
Therefore, an expression equivalent to sec(theta) sin(theta) cot(theta) csc(theta) is simply cos(theta).