Question

The value of \[\frac{\cos^3 20°- \cos^3 70°}{\sin^3 70° - \sin^3 20°}\] 

पर्याय

• \[\frac{1}{2}\]

• \[\frac{1}{\sqrt{2}}\]

•  1

• 2 

Answer 1

We have to evaluate the value. The formula to be used, 

`a^3+b^3=(a+b)(a^2+b^2-ab)` 

`a^3-b^3=(a-b)(a^2+b^2+ab)` 

So, 

=`(cos^3 20°-cos 70)/(sin^3 70°-sin^3 20)`

=` ((cos 20°-cos 70)(cos^2 20°+cos^2 70+cos 20° cos 70°))/((sin 70°-sin 20°)(sin^2 70°+sin^2 20°+sin 70° sin 20°))`

Now using the properties of complementary angles, 

= `((sin 70°- sin  20°)(sin^2 70°+cos^2 70+cos 20° cos 70°))/((sin 70°-sin 20°)(sin ^2 70°+cos ^2 70°+sin 70° sin 20°))`

=`(1+cos 20° cos 70°)/(1+sin 70° sin 20°)` 

=`( 1+ cos20° cos 70°)/(1+cos 20° cos 70°)`

=1