Question

Solve:

$\frac{\sin (-{660}^o).\tan \left({1050}^o\right).\sec (-{420}^o)}{\cos \left({225}^o\right).\csc ({315}^o.\cos ({510}^o\left)\right)}$

Answer 1

We have,

$\frac{\sin (-{660}^o).\tan \left({1050}^o\right).\sec (-{420}^o)}{\cos \left({225}^o\right).\csc ({315}^o.\cos ({510}^o\left)\right)}$

We know that

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

$\sec (-\theta )=\sec \theta$

Therefore,

$\Rightarrow -\frac{\sin \left({660}^o\right).\tan \left({1050}^o\right).\sec \left({420}^o\right)}{\cos \left({225}^o\right).\csc \left({315}^o\right).\cos \left({510}^o\right)}$

Since,

$\sin \theta =\frac{1}{\csc \theta }$

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

$\tan \theta =\frac{\sin \theta }{\cos \theta }$

So,

$\Rightarrow -\frac{\sin \left({660}^o\right).\sin \left({1050}^o\right).\sin \left({315}^o\right)}{\cos \left({225}^o\right).\cos \left({1050}^0\right)\cos \left({420}^o\right).\cos \left({510}^o\right)}$

$\Rightarrow -\frac{\sin ({360}^0+{300}^o).\sin ({720}^o+{330}^0).\sin ({360}^0-{45}^o)}{\cos ({180}^0+{45}^o).\cos ({720}^0+{330}^0)\cos (360+{60}^o).\cos ({360}^0+{150}^o)}$

$\Rightarrow -\frac{\sin \left({300}^o\right).\sin \left({330}^0\right).(-\sin ({45}^o\left)\right)}{(-\cos ({45}^o\left)\right).\cos \left({330}^0\right)\cos \left({60}^o\right).\cos \left({150}^o\right)}$

$\Rightarrow -\frac{\sin ({360}^0-{60}^o).\sin (360-{30}^0).\left(\sin \right({45}^o\left)\right)}{\left(\cos \right({45}^o\left)\right).\cos ({360}^0-{30}^0)\cos \left({60}^o\right).\cos (90+{60}^o)}$

$\Rightarrow -\frac{(-\sin ({60}^o\left)\right).(-\sin ({30}^0\left)\right).\left(\sin \right({45}^o\left)\right)}{\left(\cos \right({45}^o\left)\right).\cos \left({30}^0\right)\cos \left({60}^o\right).(-\cos ({60}^o\left)\right)}$

$\Rightarrow \frac{\left(\sin \right({60}^o\left)\right).\left(\sin \right({30}^0\left)\right).\left(\sin \right({45}^o\left)\right)}{\left(\cos \right({45}^o\left)\right).\cos \left({30}^0\right)\cos \left({60}^o\right).\left(\cos \right({60}^o\left)\right)}$

$\Rightarrow \frac{\frac{\sqrt{3}}{2}\times \frac{1}{2}\times \frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}\times \frac{\sqrt{3}}{2}\times \frac{1}{2}\times \frac{1}{2}}$

$\Rightarrow 2$

Hence, this is the answer.