2.

Asked by Premnerusu37 | 2nd Aug, 2020, 07:32: AM

Expert Answer:

If (sin-1x)2 - (cos-1x)2 = lambda pi2 and the range of lambda is [a, b] then a+b=
 
open parentheses sin to the power of negative 1 end exponent x close parentheses squared minus open parentheses cos to the power of negative 1 end exponent x close parentheses squared equals lambda straight pi squared
open parentheses sin to the power of negative 1 end exponent straight x plus cos to the power of negative 1 end exponent straight x close parentheses open parentheses sin to the power of negative 1 end exponent straight x minus cos to the power of negative 1 end exponent straight x close parentheses equals λπ squared
straight pi over 2 open parentheses straight pi over 2 minus cos to the power of negative 1 end exponent straight x minus cos to the power of negative 1 end exponent straight x close parentheses equals λπ squared
open parentheses straight pi over 2 minus 2 cos to the power of negative 1 end exponent straight x close parentheses equals 2 λπ
Since space 0 less or equal than cos to the power of negative 1 end exponent straight x less or equal than straight pi
rightwards double arrow 0 less or equal than 2 cos to the power of negative 1 end exponent straight x less or equal than 2 straight pi
rightwards double arrow 0 greater or equal than negative 2 cos to the power of negative 1 end exponent straight x greater or equal than negative 2 straight pi
rightwards double arrow straight pi over 2 greater or equal than straight pi over 2 minus 2 cos to the power of negative 1 end exponent straight x greater or equal than negative fraction numerator 3 straight pi over denominator 2 end fraction
rightwards double arrow straight pi over 2 greater or equal than 2 λπ greater or equal than negative fraction numerator 3 straight pi over denominator 2 end fraction
rightwards double arrow negative fraction numerator 3 straight pi over denominator 2 end fraction less or equal than 2 λπ less or equal than straight pi over 2
rightwards double arrow negative 3 over 4 less or equal than straight lambda less or equal than 1 fourth
Range space of space open square brackets negative 3 over 4 comma 1 fourth close square brackets
straight a plus straight b equals negative 3 over 4 plus 1 fourth equals fraction numerator negative 2 over denominator 4 end fraction equals negative 1 half

Answered by Renu Varma | 4th Aug, 2020, 12:49: PM

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