Question

Cos12°cos14cos15 value of

Answer 1

Answer:

hope this will help you

math]\cos 12°+\cos 84°+\cos 156°+\cos 132°[/math]

[math]= \cos 12°+\cos (90°-6°)+\cos (180°-24°)+\cos (90°+42°)[/math]

[math]= \cos 12°+\sin 6°-\cos 24°-\sin 42°[/math]

[math]= (\cos 12°-\cos 24°)+(\sin 6°-\sin 42°)[/math]

[math]= 2\sin 18° \sin 6° -2\cos 24° \sin 18°[/math]

[math]= 2\sin 18°(\sin 6° -\sin 66°)\quad\left[\because\cos 24° =\sin 66°.\right][/math]

[math]= 2\sin 18°(-2\sin 30° \cos 36°)[/math]

[math]=-4\sin 18°\sin 30°\cos 36° \qquad …(1)[/math]

Now, we know

[math]\sin 18° = \frac{\sqrt{5}–1}{4}, \cos 36° = \frac{\sqrt{5}+1}{4 }\quad \&\quad \sin 30° = \frac{1}{2}.[/math]

Substituting these values in (1), we get

[math]\cos 12°+\cos 84°+\cos 156°+\cos 132°[/math]

[math]= -4\times\frac{\sqrt{5}–1}{4}\times \frac{1}{2}\times \frac{\sqrt{5}+1}{4}[/math]

[math]\boxed{\boxed{=-\frac{1}{2} .}}\quad \dagger [/math]

Thank You!