Using the principal values of the inverse trigonometric functions the sum of the maximum and the minimum values of 16((sec–1x)2 + (cosec–1x)2) is:
Explanation
y=(sec−1x)2+(cosec−1x)2
=(sec−1x+cosec−1x)2−2sec−1xcosec−1x
=(2π)2−2sec−1x(2π−sec−1x)
=4π2+2(sec−1x)2−πsec−1x
=2((sec−1x)2−2πsec−1x+8π2)
=2((sec−1x−4π)2+8π2−16π2)
y=2(sec−1x−4π)2+8π2
∵ sec−1x∈[0,π]−{2π}
∴ Minimum value of y=0+8π2
Maximum value of y=2(169π2)+8π2=810π2
∴ Sum required=16(8π2+810π2)=22π2
Explanation
y=(sec−1x)2+(cosec−1x)2
=(sec−1x+cosec−1x)2−2sec−1xcosec−1x
=(2π)2−2sec−1x(2π−sec−1x)
=4π2+2(sec−1x)2−πsec−1x
=2((sec−1x)2−2πsec−1x+8π2)
=2((sec−1x−4π)2+8π2−16π2)
y=2(sec−1x−4π)2+8π2
∵ sec−1x∈[0,π]−{2π}
∴ Minimum value of y=0+8π2
Maximum value of y=2(169π2)+8π2=810π2
∴ Sum required=16(8π2+810π2)=22π2
