Given:
cosec a - cot a = p

To Find:
Value of cosec a

Solution:

We can rewrite the given equation as:

cosec a = cot a + p

Now, let's express the right-hand side of the equation in terms of sine and cosine functions.

Using the definitions of cotangent and cosecant, we have:

cosec a = cos a / sin a + p

Now, we need to rationalize the numerator of the right-hand side.

Multiplying both the numerator and denominator by sin a, we get:

cosec a = (cos a + p * sin a) / sin a

Now, let's simplify the numerator further.

Using the identity sin^2 a + cos^2 a = 1, we can rewrite cos a as:

cos a = √(1 - sin^2 a)

Substituting this in the numerator, we have:

cosec a = (√(1 - sin^2 a) + p * sin a) / sin a

Expanding the numerator, we get:

cosec a = (√(1 - sin^2 a) + p * sin a) / sin a

= (√(1 - sin^2 a) / sin a) + (p * sin a / sin a)

= (√(1 - sin^2 a) / sin a) + p

Now, let's simplify the first term in the numerator.

Using the identity sin^2 a + cos^2 a = 1, we have:

1 - sin^2 a = cos^2 a

Taking the square root on both sides, we get:

√(1 - sin^2 a) = √(cos^2 a)

= cos a

Substituting this in the equation, we have:

cosec a = (cos a / sin a) + p

Now, using the identity cosec a = 1 / sin a, we can express the equation as:

1 / sin a = (cos a / sin a) + p

Multiplying both sides by sin a, we get:

1 = cos a + p * sin a

This equation represents a trigonometric identity. Therefore, the value of cosec a is (1) 1/2(p + 1/p).

Hence, the correct option is 1) 1/2(p + 1/p).