Identities: Sum to Product

$\frac{cos\delta -cos7\delta }{sin\delta +sin7\delta }=tan3\delta$
 

From Sum-to-product formula for cosine
$cos\alpha +cos\beta =-2 sin\left(\frac{\alpha +\beta }{2}\right) sin\left(\frac{\alpha -\beta }{2}\right)$
and
$\sin \alpha +sin\beta =2 sin\left(\frac{\alpha +\beta }{2}\right) cos\left(\frac{\alpha -\beta }{2}\right)$

=$\frac{-2 sin\left(\frac{\delta +7\delta }{2}\right) sin\left(\frac{\delta -7\delta }{2}\right)}{2sin\left(\frac{\delta +7\delta }{2}\right)cos\left(\frac{\delta -7\delta }{2}\right)}$

=$\frac{-2 sin\left(\frac{8\delta }{2}\right) sin\left(\frac{-6\delta }{2}\right)}{2sin\left(\frac{8\delta }{2}\right)cos\left(\frac{-6\delta }{2}\right)}$

=$\frac{-2 sin\left(\frac{\cancel{8}\delta }{\cancel{2}}\right) sin\left(\frac{\cancel{-6}\delta }{\cancel{2}}\right)}{2sin\left(\frac{\cancel{8}\delta }{\cancel{2}}\right)cos\left(\frac{\cancel{-6}\delta }{\cancel{2}}\right)}$

=$\frac{-2 sin\left(4\delta \right) sin(-3\delta )}{2sin\left(4\delta \right)cos(-3\delta )}$

=$\frac{-\cancel{2sin\left(4\delta \right)} sin(-3\delta )}{\cancel{2sin\left(4\delta \right)}cos(-3\delta )}$

=$\frac{-sin(-3\delta )}{cos(-3\delta )}$

=$\frac{sin3\delta }{cos(-3\delta )}$

Note: By odd/even trigonometric function identities
Even Trigonometric Function Identities:

1. Cosine is an even function; $\cos (-x)=cos\left(x\right)$
2. Secant is an even function;$\sec (-x)=sec\left(x\right)$

Odd Trigonometric Function Identities:

 

1. Sine is an odd function;$\sin (-x)=-sin\left(x\right)$
2. Cosecant is an odd function; $\csc (-x)=-csc\left(x\right)$
3. Tangent is an odd function;$\tan (-x)=-tan\left(x\right)$

=$\frac{sin3\delta }{cos3\delta }$

=$\tan 3\delta$