Question

prove that cos35°+cos85°-sin65°=0 ..
please help me to solve it

Answer 1

Now we know that
$\cos( \alpha ) + \cos( \beta ) = 2 \cos( \frac{ \alpha + \beta }{2} ) \cos( \frac{ \alpha - \beta }{2} )$
Now consider this
$\cos(35) + \cos(85) \\ = 2 \cos( \frac{35 + 85}{2} ) \cos( \frac{85 - 35}{2} ) \\ = 2 \cos( \frac{120}{2} ) \cos( \frac{50}{2} ) \\ = 2 \cos(60) \cos(25) \\ = 2 \times \frac{1}{2} \times \cos(25) \\ = \cos(25) \\ = \sin(90 - 25) = \sin(65)$
This gives
$\cos(35) + \cos(85) = \sin(65) \\ \cos(35) + \cos(85) - \sin(65) = 0$