Question

Prove that sin x - sin y / cos x + cos y = tan (x-y/2)​

Answer 1

We know the trigonometric identities,

$\sin x-\sin y=2\cos\left (\dfrac {x+y}{2}\right)\sin\left (\dfrac {x-y}{2}\right)$

And,

$\cos x+\cos y=2\cos\left (\dfrac {x+y}{2}\right)\cos\left (\dfrac {x-y}{2}\right)$

Thus,

$\textsf{LHS}\\\\\\=\dfrac{\sin x-\sin y}{\cos x+\cos y}\\\\\\=\dfrac{2\cos\left (\dfrac {x+y}{2}\right)\sin\left (\dfrac {x-y}{2}\right)}{2\cos\left (\dfrac {x+y}{2}\right)\cos\left (\dfrac {x-y}{2}\right)}\\\\\\=\dfrac{\sin\left (\dfrac {x-y}{2}\right)}{\cos\left (\dfrac {x-y}{2}\right)}\\\\\\=\tan\left (\dfrac{x-y}{2}\right)\\\\\\=\textsf {RHS}$

Hence Proved!