tan2x=(2tanx)/(1-tan^2x)

Use the compound angle formula for the tan function.

I hope you know it, right?

tan(2x)=\frac{2tan(x)}{1-tan^{2}(x)}

When proving trig identities, try getting into the habit of converting everything to sin and cos.

\frac{sin(2x)}{cos(2x)}=\frac{\frac{2sin(x)}{cos(x )}}{1-\frac{sin^{2}(x)}{cos^{2}(x)}}

The right side becomes:

\frac{\frac{2sin(x)}{cos(x)}}{\frac{cos^{2}(x)-sin^{2}(x)}{cos^{2}(x)}}

\frac{2sin(x)}{\not{c}\not{o}\not{s} \not{x}} \cdot \frac{cos^{\not{2}^{1}}(x)}{cos^{2}(x)-sin^{2}(x)}

Now, knowing that cos^{2}(x)-sin^{2}(x)=cos(2x)

and sin(2x)=2sin(x)cos(x)

we get:

2sin(x)\cdot\frac{cos(x)}{cos(2x)}

\frac{sin(2x)}{cos(2x)}=tan(2x)

Cos^2x+sin^2x/2sinxcosx

(sin(X))/(tan(X))+(cos(X))/(cot(X))=sin(X)+cos(X)

cos^2x=(1/2)cos2x

cos^2x=(1/2)cos2x

Answered here:
https://www.askmehelpdesk.com/mathematics/verify-identity-cos-2x-1-2-cos2x-650298.html

(sin(X))/(tan(X))+(cos(X))/(cot(X))=sin(X)+cos(X)

Simply convert the tan(x) and cot(x) functions to their sine and cosine equivalents, and it comes right out.

cot 2x= cot^2x-1/2cotx

cot 2x= cot^2x-1/2cotx

No, the identity is incorrect: cot(2x) \ \ne \ cot^2(x) - \frac 12 cot(x)