Question

If cosecθ+cotθ=m and cosecθ-cotθ=n, prove that mn=1.

Answer 1

Answer:

mn=1

Step-by-step explanation:

1. (cosecθ+cotθ)=m

(cosecθ-cotθ)=n

multiplying both the equations,

(cosecθ+cotθ)(cosecθ-cotθ)=m×n

cosec²θ-cot²θ=m×n

using the identity: 1+cot²θ=cosec²θ

m×n=1

Answer 2

If $\csc \theta+cot \theta$ = m and $\csc \theta-cot \theta$ = n, then mn = 1, proved.

Step-by-step explanation:

We have,

$\csc \theta+cot \theta$ = m           ............. (1)

and

$\csc \theta-cot \theta$ = n           ............. (2)

To prove that, mn = 1.

Multiplying equations (1) and (2), we get

$(\csc \theta+cot \theta)(\csc \theta-cot \theta)$ = mn

Using the algebraic identity,

$a^{2} -b^{2}$ = (a + b)(a -b)

⇒ $\csc^2 \theta-cot^2 \theta$ = mn

Using the trigonometric identity,

$\csc^2 A-cot^2 A$ = 1

⇒ 1 = mn

⇒ mn = 1, proved.

Thus, if $\csc \theta+cot \theta$ = m and $\csc \theta-cot \theta$ = n, then mn = 1, proved.