Question

Verify the identity pre calculus

Answer 1

Answer:

The verification is in the explanation.

Step-by-step explanation:

I'm going to start with the left and play with it til I get the right.

First step: Expand the binomial square using $(u+v)^2=u^2+2uv+v^2$.

$(\tan(x)+\cot(x))^2$

$\tan^2(x)+2\tan(x)\cot(x)+\cot^2(x)$

Second step: Realize $\tan(x)$ and $cot(x)$ are reciprocal functions. Their product will be 1.

[Side note: $\tan(x)\cot(x)=\frac{\sin(x}{\cos(x)}\frac{\cos(x)}{sin(x)}=\frac{\sin(x)\cos(x)}{\sin(x)\cos(x)}=1$.]

$\tan^2(x)+2(1)+\cot^2(x)$

$\tan^2(x)+2+\cot^2(x)$

Third step: Recall the Pythagorean Identities $\tan^2(x)+1=sec^2(x)$ and $1+cot^2(x)=csc^2(x)$:

$\tan^2(x)+(1+1)+\cot^2(x)$

$\tan^2(x)+1+1+\cot^2(x)$

$(\tan^2(x)+1)+(1+\cot^2(x))$

$\sec^2(x)+\csc^2(x)$