Aim to find from the Howe-Tucker action: $$S_{\text{HT}}=-\frac{1}{2}\int d^d\sigma\sqrt{-\gamma}(\gamma^{ab}\partial_a X^{\mu}\partial_b X^{\nu}\eta_{\mu\nu}-m^2(d-2))$$ (which is a Polyakov-like action), the Nambu-Goto form of the action: $$S_{\text{NG}}=-m^{d-2}\int d^d\sigma\sqrt{-det(\partial_a X^{\mu}\partial_b X^{\nu}\eta_{\mu\nu})}.$$
I figured for the induced relations that: $$\gamma^{ab}\delta\gamma_{ab}=2\gamma^{ab}\partial_a X^{\mu}\partial_b\delta X^{\nu}$$ $$\delta\sqrt{-\gamma}=\gamma^{ab}\delta\gamma_{ab}.$$
I have tried to find the equation of motion so that I may substitute it back in. However, I get: $$-\frac{1}{2}\gamma^{ab}\partial_a X^{\mu}\partial_bX^{\nu}\eta_{\mu\nu}+\frac{1}{4}m^2(d-2)=0$$ when I should have $$-\frac{1}{2}\partial_a X^{\mu}\partial_bX^{\nu}\eta_{\mu\nu}+\frac{1}{4}\gamma_{ab}(\gamma^{cd}\partial_c X^{\mu}\partial_dX^{\nu}\eta_{\mu\nu}-m^2(d-2)).$$
Any suggestions on where I may be going wrong would be appreciated.
