Let's solve the given expression step by step:
The expression is:
| √3cos10∘−sin10∘ |
| sin25∘cos25∘ |
First, we need to simplify the denominator. Recall that:
sin25∘cos25∘=sin50∘Thus, the expression becomes:
| √3cos10∘−sin10∘ |
| sin50∘ |
=2| √3cos10∘−sin10∘ |
| sin50∘ |
Next, let's consider the numerator. Note that:
sin50∘=cos(90∘−50∘)=cos40∘Thus, we need to express the numerator in terms of angles close to 40 degrees. We also know that:
cos40∘=cos(180∘−140∘)=−cos140∘Let's use the identity for the cosine of a sum:
cos40∘=cos(10∘+30∘)=cos10∘cos30∘−sin10∘sin30∘We know that:
cos30∘= and sin30∘= So:
cos40∘=cos10∘⋅−sin10∘⋅⇒2cos40∘=√3cos10∘−sin10∘This means our numerator simplifies to:
2cos40∘Thus the entire expression becomes:
2=2⋅2=4