If 𝑐𝑜𝑠 (α + β) = 0, then 𝑠𝑖𝑛 (α − β) can be reduced to

We have been given that 𝑐𝑜𝑠 (α + β) = 0. 

We know that 𝑐𝑜𝑠 90° = 0

⇒ 𝑐𝑜𝑠 (α + β) = 𝑐𝑜𝑠 90°

⇒ α + β = 90°

⇒ α = 90° − β

Substituting the value of α from the above equation, we get 

Now, 𝑠𝑖𝑛 (α − β) = 𝑠𝑖𝑛 (90° − β − β)

⇒ 𝑠𝑖𝑛 (α − β) = 𝑠𝑖𝑛 (90° − 2β)

We know that, for any angle θ, 𝑠𝑖𝑛 (90° − θ) = 𝑐𝑜𝑠 θ 

Therefore we can say that, 𝑠𝑖𝑛 (90° − 2β) = 𝑐𝑜𝑠 2β 

Hence, here 𝑠𝑖𝑛 (α − β) can be reduced to 𝑐𝑜𝑠 2β.