If a cot θ + b cosec θ = p and b cot θ + a cosec θ = q, then p² – q² =

We are given the equations:
a cotθ + b cosecθ = p — (1)
b cotθ + a cosecθ = q — (2)

We need to find the value of p² – q².

Step 1: Recall the identity for p² – q²
The difference of squares formula states:
p² – q² = (p + q)(p – q).

Thus, we first calculate p + q and p – q.

Step 2: Add equations (1) and (2)
Add the two equations:
(a cotθ + b cosecθ) + (b cotθ + a cosecθ) = p + q.

Simplify:
(a + b)cotθ + (a + b)cosecθ = p + q.

Factor out (a + b):
(a + b)(cotθ + cosecθ) = p + q. — (3)

Step 3: Subtract equation (2) from (1)
Subtract equation (2) from equation (1):
(a cotθ + b cosecθ) – (b cotθ + a cosecθ) = p – q.

Simplify:
(a – b)cotθ + (b – a)cosecθ = p – q.

Factor out (a – b):
(a – b)(cotθ – cosecθ) = p – q. — (4)

Step 4: Substitute into p² – q²
Using the identity p² – q² = (p + q)(p – q), substitute the expressions for p + q and p – q from equations (3) and (4):
p² – q² = [(a + b)(cotθ + cosecθ)] × [(a – b)(cotθ – cosecθ)].

Simplify:
p² – q² = (a + b)(a – b) × (cotθ + cosecθ)(cotθ – cosecθ).

Step 5: Simplify further using the difference of squares
The term (cotθ + cosecθ)(cotθ – cosecθ) is a difference of squares:
(cotθ + cosecθ)(cotθ – cosecθ) = cot²θ – cosec²θ.

From the trigonometric identity:
cosec²θ – cot²θ = 1,
we can write:
cot²θ – cosec²θ = -1.

Substitute this into the expression for p² – q²:
p² – q² = (a + b)(a – b) × (-1).

Simplify:
p² – q² = -(a + b)(a – b).

Expand (a + b)(a – b) using the difference of squares:
(a + b)(a – b) = a² – b².

Thus:
p² – q² = -(a² – b²).
This question is related to Chapter 8, “Introduction to Trigonometry,” from the Class 10th NCERT Mathematics textbook. Answer the question using your knowledge and understanding of the chapter.

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