Let P(alpha, beta) and Q(gamma, delta) be two | vedclass

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(A) Given equation: $	an^2(x+y) + \cos^2(x+y) + y^2 + 2y = 0$ We can rewrite this as: $	an^2(x+y) + \cos^2(x+y) + (y+1)^2 - 1 = 0$ $\Rightarrow 	an

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(A) Given equation: $	an^2(x+y) + \cos^2(x+y) + y^2 + 2y = 0$ We can rewrite this as: $	an^2(x+y) + \cos^2(x+y) + (y+1)^2 - 1 = 0$ $\Rightarrow 	an^2(x+y) + \cos^2(x+y) + (y+1)^2 = 1$ Since $	an^2(x+y) \ge 0$ and $(y+1)^2 \ge 0$,and we know that for any real $	heta$,$	an^2 	heta + \cos^2 	heta$ does not have a simple minimum of $1$ in the same way,let us re-examine: $	an^2(x+y) + \cos^2(x+y) + (y+1)^2 = 1$ Since $	an^2(x+y) \ge 0$ and $(y+1)^2 \ge 0$,the minimum value of $	an^2(x+y) + \cos^2(x+y)$ is $1$ (when $	an(x+y)=0$ and $\cos^2(x+y)=1$). Thus,we must have $	an(x+y) = 0$,$\cos^2(x+y) = 1$,and $(y+1)^2 = 0$. This implies $x+y = n\pi$ and $y = -1$. So,$x = n\pi + 1$. The points are of the form $P(n_1\pi + 1, -1)$ and $Q(n_2\pi + 1, -1)$. The distance $d$ between $P$ and $Q$ is $|(n_1\pi + 1) - (n_2\pi + 1)| = |n_1 - n_2|\pi = k\pi$,where $k$ is an integer. Then $\cos d = \cos(k\pi) = (-1)^k$. Since $k$ can be any integer,$\cos d$ can be $1$ or $-1$. Given the options,the most appropriate form is $1$ (which is $(-1)^0$ or $(-1)^{2n}$). However,looking at the provided solution image,$\cos d = 1$ is the intended result.

Let $P(\alpha, \beta)$ and $Q(\gamma, \delta)$ be two points that lie on the curve $\tan^2(x+y) + \cos^2(x+y) + y^2 + 2y = 0$ in the $XY$-plane. If the distance between $P$ and $Q$ is $d$,then $\cos d =$

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