Question

Evaluate:
$\frac{3\cos {55}^{\circ }}{7\sin {35}^{\circ }}-\frac{4\left(\cos {70}^{\circ }\csc {20}^{\circ }\right)}{7\left(\tan 5^{\circ }\tan {25}^{\circ }\tan {45}^{\circ }\tan {65}^{\circ }\tan {85}^{\circ }\right)}$

Answer 1

We know,

$\cos (90-\theta )=\sin \theta$

$\tan (90-\theta )=\cot \theta$

$\cot \theta =\frac{1}{\tan \theta }$

$\sec \theta =\frac{1}{\cos \theta }$

$\begin{array}{cc}\cos {55}^{\circ }& =\cos ({90}^{\circ }-{35}^{\circ })\\ & =\sin {35}^{\circ }\\ \tan 5^{\circ }& =\tan ({90}^{\circ }-{85}^{\circ })\\ & =\cot {85}^{\circ }\\ \tan {25}^{\circ }& =\tan ({90}^{\circ }-{65}^{\circ })\\ & =\cot {65}^{\circ }\\ \csc {20}^{\circ }& =\sec ({90}^{\circ }-{70}^{\circ })\\ & =\sec {70}^{\circ }\end{array}$

Now we will use the above identities to find the value of given expression,

$\frac{3\cos {55}^{\circ }}{7\sin {35}^{\circ }}-\frac{4\left(\cos {70}^{\circ }\csc {20}^{\circ }\right)}{7\left(\tan 5^{\circ }\tan {25}^{\circ }\tan {45}^{\circ }\tan {65}^{\circ }\tan {85}^{\circ }\right)}$

$\Rightarrow \frac{3\sin {35}^{\circ }}{7\sin {35}^{\circ }}-\frac{4\left(\cos {70}^{\circ }\sec {70}^{\circ }\right)}{7\left(\tan 5^{\circ }\tan {25}^{\circ }\tan {45}^{\circ }\cot {25}^{\circ }\cot 5^{\circ }\right)}$

$\Rightarrow \frac{3}{7}-\frac{4\left(\cos {70}^{\circ }\tan 5^{\circ }\tan {25}^{\circ }\right)}{7\left(\tan {45}^{\circ }\tan {25}^{\circ }\tan 5^{\circ }\cos {70}^{\circ }\right)}$

$\Rightarrow \frac{3}{7}-\frac{4}{7}$

$\Rightarrow -\frac{1}{7}$