The correct option is C Exactly one value of x exists
Given that 1−2x−x2=tan2(x+y)+cot2(x+y)
Using a2+b2=(a−b)2+2ab
⇒1−2x−x2=[tan(x+y)−cot(x+y)]2+2tan(x+y)cot(x+y)
⇒1−2x−x2=[tan(x+y)−cot(x+y)]2+2
⇒−2+1−2x−x2=[tan(x+y)−cot(x+y)]2
⇒−1−2x−x2=[tan(x+y)−cot(x+y)]2
⇒−(x+1)2=[tan(x+y)−cot(x+y)]2
⇒(x+1)2+[tan(x+y)−cot(x+y)]2=0
⇒(x+1)2=0 & [tan(x+y)−cot(x+y)]2=0
⇒x=−1 & tan(x+y)=cot(x+y)
⇒tan2(x+y)=1
tan2(x+y)=tan2π4
⇒x+y=nπ±π4,n∈Z
⇒−1+y=nπ±π4,n∈Z
⇒y=1+nπ±π4,n∈Z