Question

What is the value of $(\frac{1}{\sin {45}^0}-\sin {45}^0)(\frac{1}{\cos {45}^0}-\cos {45}^0)(\tan {45}^0+\frac{1}{\tan {45}^0})$ ?

Answer 1

The given expression has the value
$(\frac{1}{\frac{1}{\sqrt{2}}}-\frac{1}{\sqrt{2}})(\frac{1}{\frac{1}{\sqrt{2}}}-\frac{1}{\sqrt{2}})(1+\frac{1}{1})$
$=(\sqrt{2}-\frac{1}{\sqrt{2}})(\sqrt{2}-\frac{1}{\sqrt{2}})\times 2$
$=\frac{2-1}{\sqrt{2}}\times \frac{2-1}{\sqrt{2}}\times 2$
$=\frac{1}{\sqrt{2}}\times \frac{1}{\sqrt{2}}\times 2=1$