Given, (cos 8A * cos 5A - cos 12A * cos 9A)/(sin 8A * cos 5A + cos 12A * sin 9A)

= {2(cos 8A * cos 5A - cos 12A * cos9A)/2}/{2(sin 8A * cos 5A + cos 12A * sin 9A)/2}                    {Multiply and divide by 2}

= [{cos 13A + cos 3A}/2 - {cos 21A + cos 3A}/2]/[{sin 13A + sin 3A}/2 + {sin 21A - sin 3A}/2]      {Apply product into sum or difference formula of

                                                                                                                                                             trigonometric ratios}

= (cos 13A - cos 21A)/(sin 13A + sin 21A)                                                    

= (2*sin 17A * sin 4A)/(2*sin 17A * cos4A)                        {Apply sum and difference into product formula of trigonometric ratios}

= sin 4A/cos 4A

= tan 4A

So, (cos 8A * cos 5A - cos 12A * cos 9A)/(sin 8A * cos 5A + cos 12A * sin 9A) = tan 4A