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प्रश्न
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
उत्तर
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
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