Question

True or false (Picture provided)

Answer 1

Hello!

The answer is: True.

Why?

A counterexample is a way that we can prove that something is not true about a mathematical equation or expression, it's also considered as an exception to a rule.

So:

$sec^{2}x-1=\frac{cosx}{cscx}\\\\tg^{2}x=\frac{cosx}{\frac{1}{sinx}}\\\\\frac{1-cos2x}{1+cos2x}=cosxsinx$

Then, evaluating we have:

$\frac{1-cos(2*45)}{1+cos(2*45)}=cos(45)*sin(45)\\\\\frac{1-0}{1+0}=\frac{\sqrt{2} }{2}*\frac{\sqrt{2}}{2}\\\\1=\frac{(\sqrt{2})^{2} }{4}\\\\1=\frac{2}{4}\\\\1=\frac{1}{2}$

Hence, we can see that the equation is not fulfilled, so, 45° is a counterexample for $sec^{2}x-1=\frac{cosx}{cscx}$ and the answer is true.

Have a nice day!