Problem: If sec(theta) * tan(theta) = p, find the value of cosec(theta).

Solution:

To find the value of cosec(theta), we need to manipulate the given equation involving sec(theta) and tan(theta).

Step 1: Rewrite sec(theta) and tan(theta) in terms of sine and cosine.

Recall the definitions of sec(theta) and tan(theta):

sec(theta) = 1/cos(theta)
tan(theta) = sin(theta)/cos(theta)

Substituting these definitions into the given equation, we get:

(1/cos(theta)) * (sin(theta)/cos(theta)) = p

Simplifying further, we have:

sin(theta)/cos^2(theta) = p

Step 2: Rewrite the equation using the reciprocal identities for sine and cosine.

The reciprocal identities state:

cosec(theta) = 1/sin(theta)
sec^2(theta) = 1/cos^2(theta)

Using these identities, we can rewrite the equation as:

cosec(theta) * sec^2(theta) = p

Step 3: Substitute the value of sec^2(theta) using the Pythagorean identity.

The Pythagorean identity states:

sin^2(theta) + cos^2(theta) = 1

Rearranging this equation, we have:

cos^2(theta) = 1 - sin^2(theta)

Substituting this into our equation, we get:

cosec(theta) * (1 - sin^2(theta)) = p

Step 4: Simplify the equation and solve for cosec(theta).

Expanding the equation, we get:

cosec(theta) - cosec(theta) * sin^2(theta) = p

Rearranging the terms, we have:

cosec(theta) = p / (1 - sin^2(theta))

Using the identity cos^2(theta) = 1 - sin^2(theta), we can simplify further:

cosec(theta) = p / cos^2(theta)

Step 5: Substitute the value of cos^2(theta) using the reciprocal identity.

The reciprocal identity states:

cos(theta) = 1/sec(theta)

Substituting this into our equation, we get:

cosec(theta) = p / (1/sec^2(theta))

Simplifying further, we have:

cosec(theta) = p * sec^2(theta)

Step 6: Substitute the value of sec^2(theta) using the reciprocal identity.

The reciprocal identity states:

sec(theta) = 1/cos(theta)

Substituting this into our equation, we get:

cosec(theta) = p * (1/cos^2(theta))

Simplifying further, we have:

cosec(theta) = p / cos^2(theta)

Step 7: Substitute the value of cos^2(theta) using the Pythagorean identity.

Using the Pythagorean identity, cos^2(theta) = 1 - sin^2(theta), we can rewrite the equation as:

cosec(theta) = p / (1 - sin^2(theta))

Conclusion:

The value of cosec(theta) is p / (1 - sin^2(theta)).