cos ( alpha + beta ) sin ( gamma + delta ) = cos ( alpha - beta ) sin - Maths - Trigonometric Functions

Question

cos ( alpha + beta ) sin ( gamma + delta ) = cos ( alpha - beta
) sin ( gamma + delta ) then prove that cot alpha cot beta cot
gamma = cot delta

Ajanta Trivedi · Accepted Answer

the given equation is: $\cos (α + β) \sin (γ + δ) = \cos (α - β) \sin (γ - δ)$ .............(1)

i.e.

$\frac{\cos (α + β)}{\cos (α - β)} = \frac{\sin (γ - δ)}{\sin (γ + δ)}$

now applying componendo and dividendo:

$\frac{\cos (α + β) + \cos (α - β)}{\cos (α + β) - \cos (α - β)} = \frac{\sin (γ - δ) + \sin (γ + δ)}{\sin (γ - δ) - \sin (γ + δ)} \frac{2 \cos α \cdot \cos β}{2 \sin α \cdot \sin β} = \frac{2 \sin γ \cdot \cos δ}{2 \cos γ \cdot \sin δ} \frac{\cos α \cdot \cos β}{\sin α \cdot \sin β} = \frac{\sin γ \cdot \cos δ}{\cos γ \cdot \sin δ} \Rightarrow \cot α \cdot \cot β = \frac{\cot δ}{\cot γ}$

$\Rightarrow \cot α \cdot \cot β \cdot \cot γ = \cot δ$

which is the required result.

hope this helps you.

cheers!!

cos ( alpha + beta ) sin ( gamma + delta ) = cos ( alpha - beta ) sin ( gamma + delta ) then prove that cot alpha cot beta cot gamma = cot delta