Is there no such identity as $\csc^2+\sec^2=1$? $$\csc^2+\sec^2=1?$$

I thought I could just use reciprocal from the other formula $\sin^2+\cos^2=1$, can you explain what's wrong?
You can't just "use reciprocal", as you put it, because
$$\frac{1}{x^2+y^2}$$
is not the same thing as
$$\frac{1}{x^2}+\frac{1}{y^2}$$
We have $sin^{2}x + cos^{2}x = 1$ .Taking reciprocal we get $1 / sin^{2}x + cos^{2}x = 1$ ,then converting sin to cosec and cos to sec and then taking lcm of denominator we get $sec^{2}xcosec^{2}x/sec^{2}x + cosec^{2}x$ = 1