(sin(x)cos(x))/(cos^2(x))-(sin^2(x))=(tan(x))/(1-tan^2(x))

Question

dddddddd dddddddddd ddddddddddddd

Answer 1

sin(x)cos(x)=sin(2x)/2; cos²(x)-sin²(x)=cos(2x) are standard identities.

So on the left side we therefore have sin(2x)/2cos(2x)=tan(2x)/2 by definition of tangent.

tan(a+b)=(tan(a)+tan(b))/(1-tan(a)tan(b)) is another standard identity. When a=b=x:

tan(2x)=2tan(x)/(1-tan²(x)), and we have tan(2x)/2=tan(x)/(1-tan²(x)), which is the right side.

Therefore: sin(x)cos(x)/(cos²(x)-sin²(x))=tan(x)/(1-tan²(x)) QED