Question

Prove that cos15 - sin15/cos15 + sin15=1/root 3

Answer 1

sin(15) + cos(15) / (sin(15) - cos(15)
sin(15) + cos(15) * (sin(15) + cos(15) / (sin(15) - cos(15) * sin(15) + cos(15)
(sin(15)^2 + 2sin(15)cos(15) + cos(15)^2) / (sin(15)^2 - cos(15)^2
sin(15)^2 + cos(15)^2 + 2sin(15) cos (15) / cos(15)^2 - sin(15)^2
1 + sin(2 * 15) / cos(2 * 15)
1 + sin(30)) / cos(30) 
1 + 1/2) / (3)/2)
(3/2) / √(3)/2) 

√ 3/(3)
√(3)

Answer 2

Answer:

To Prove : $\frac{cos15 - sin15}{cos15 + sin15}=\frac{1}{\sqrt{3}}$

Solution :

$\frac{cos15 - sin15}{cos15 + sin15}$

$\frac{cos15 - sin15}{cos15 + sin15} \times \frac{cos15 - sin15}{cos15 - sin15}$

$\frac{(cos15 - sin15)^2}{cos^15 - sin^15}$

Identity : $(a-b)^2=a^2+b^2-2ab\\Cos^2x-Sin^2x = Cos 2x$

$\frac{cos^2 15 + sin^2 15- 2sin 15 cos 15 }{Cos 30}$

Identity : $Cos^2x +Sin^2x = 1$

$\frac{1- 2sin 15 cos 15}{Cos 30}$

Identity : $2 sin A cos A = Sin 2A$

$\frac{1-Sin30}{Cos 30}$

$\frac{1-\frac{1}{2}}{\frac{\sqrt{3}}{2}}$

$\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}$

$\frac{1}{\sqrt{3}}$

Hence proved