Question

Prove that (cos 20- sin20)divided by (cos 20 + sin 20) is equal to tan25 All in degrees Please answer as soon as possible

Answer 1

$\frac{cos 20- sin20}{cos 20 + sin 20}$

$\frac{cos 20(1- \frac{sin20}{cos20}) }{cos 20(1 + \frac{sin20}{cos20} }$

$\frac{cos 20(1- tan20) }{cos 20(1 + tan20) }$

$\frac{1- tan20 }{1 + tan20 }$

since, $\frac{1- tan\alpha }{1 + tan \alpha }= tan (45 -\alpha)$

$\frac{1- tan20 }{1 + tan20 }$ = Tan(45 - 20) = Tan 25

Hence proved

Answer 2

Answer:

Given : $\frac{cos 20 - sin 20 }{cos 20 + sin 20}$

To Find : $\frac{cos 20 - sin 20 }{cos 20 + sin 20}=tan 25$

Solution:

$\frac{cos 20 - sin 20 }{cos 20 + sin 20}$

Take cos 20 common

$\frac{cos 20(1- \frac{sin20}{cos20}) }{cos 20(1 +  \frac{sin20}{cos20} )}$

Identity : $\frac{Sin x}{cos x}=tan x$

$\frac{cos 20(1- tan20) }{cos 20(1 + tan20) }$

$\frac{1- tan20 }{1 + tan20 }$

since, $\frac{1- tan\alpha  }{1 + tan \alpha }= tan (45 -\alpha)$ ---- A

Using A

$\frac{1- tan20 }{1 + tan20 }  = Tan(45 - 20) = Tan 25$

Hence proved .