Question

is cos^2x(1+tan^2x)=1 an identity? explain your reasoning.

Answer 1

I would say so

\[\large cos^2(x)(1 + tan^2(x)) = 1\]

What is \(\large 1 + tan^2(x)\) ?

Answer 2

\[\sec^2 x\]

Answer 3

Right...so

\[\large cos^2(x)sec^2(x) = 1\]

Now...we know \(\large sec^2(x)\) can be written as \(\large \frac{1}{cos^2(x)}\)

So

\[\large cos^2(x)(\frac{1}{cos^2(x)}) = 1\]
\[\large \frac{cos^2(x)}{cos^2(x)} = 1\]

Is that a true statement?

Answer 4

yes, thank you!