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Maths-

General

Easy

Question

$left parenthesis sin space x plus cos space x right parenthesis squared plus cos squared space open parentheses pi over 4 plus x close parentheses element of$

$[0, 1]$
$[0, 2]$
$[1, 2]$
$[0, 3]$

hint Hint:

In this question, we have to find the range of the given function. For that first we will simplify the given equation and later find the range.

The correct answer is: $left square bracket 1 comma 2 right square bracket$

$left parenthesis sin space x plus cos space x right parenthesis squared plus cos squared space open parentheses pi over 4 plus x close parentheses equals left parenthesis sin space x plus cos space x right parenthesis squared plus open parentheses cos straight pi over 4. cos space x minus sin straight pi over 4. sin space x close parentheses squared equals left parenthesis sin space x plus cos space x right parenthesis squared plus open parentheses fraction numerator 1 over denominator square root of 2 end fraction cos space x minus fraction numerator 1 over denominator square root of 2 end fraction sin space x close parentheses squared equals sin squared x plus cos squared x plus 2 sin x. cos space x plus 1 half sin squared x plus 1 half cos squared x minus 1 half cross times 2 sin space x. cos x space equals 1 plus 1 half open parentheses sin squared x plus cos squared x close parentheses plus sin x. cos space x equals 3 over 2 plus fraction numerator sin space 2 x over denominator 2 end fraction equals 1 half open parentheses 3 plus sin space 2 x close parentheses minus 1 less or equal than sin space y less or equal than 1 rightwards double arrow negative 1 less or equal than sin space 2 x less or equal than 1 rightwards double arrow negative 1 plus 3 less or equal than 3 plus sin space 2 x less or equal than 1 plus 3 rightwards double arrow 2 over 2 less or equal than 1 half open parentheses 3 plus sin space 2 x close parentheses less or equal than 4 over 2 rightwards double arrow 1 less or equal than 1 half open parentheses 3 plus sin space 2 x close parentheses less or equal than 2 rightwards double arrow left parenthesis sin space x plus cos space x right parenthesis squared plus cos squared space open parentheses pi over 4 plus x close parentheses element of open square brackets 1 comma 2 close square brackets$

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