Question

sqrt((25 tan x )^2 +25). should the answer equate to sec^2 x? or 5 sec x ?

Answer 1

Well, I assume you mean this
\[ \sqrt{25\tan^2 x + 25} \]
which is equal to
\[5 \sqrt{\tan^2 x + 1} \]
because \( 5 = \sqrt{25} \).


Can you finish the problem now yourself?

Answer 2

I suppose it is worth memorizing the fact that
\[ tan^2 x+ 1 = sec^2 x \]

though it is definitely worth knowing that
\[ sin^2 x + cos^2 x =1 \]

we can use this to show the first identity. Of course, you know
\[ tan (x) = \frac{sin (x)}{cos( x)} \]
if we add 1 to tan^2(x):
\[ \frac{sin^2 (x)}{cos^2( x)} + \frac{cos^2(x)}{cos^2(x)} \]
which is the same as
\[ \frac{sin^2 (x)+cos^2( x)}{cos^2( x)} \]
now use the very important identity to make the top 1. We get
\[ \frac{1}{cos^2( x)} = sec^2(x) \]
so we now know
\[ tan^2 x+ 1 = sec^2 x \]

Answer 3

oh okay got it now thank you