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Question

The number of distinct real roots of $open vertical bar table row cell sin invisible function application x end cell cell cos invisible function application x end cell cell cos invisible function application x end cell row cell cos invisible function application x end cell cell sin invisible function application x end cell cell cos invisible function application x end cell row cell cos invisible function application x end cell cell cos invisible function application x end cell cell sin invisible function application x end cell end table close vertical bar equals 0$ in the interval $– fraction numerator pi over denominator 4 end fraction less or equal than x less or equal than fraction numerator pi over denominator 4 end fraction$ is

0
2
1
3

The correct answer is: 1

To simplify the determinant, let $sin invisible function application x equals a semicolon cos invisible function application x equals b$ . Then the equation becomes
$open vertical bar table row a b b row b a b row b b a end table close vertical bar equals 0$ . Operating $C subscript 2 end subscript rightwards arrow C subscript 2 end subscript minus C subscript 1 end subscript semicolon C subscript 3 end subscript rightwards arrow C subscript 3 end subscript minus C subscript 2 end subscript$ , we get
$open vertical bar table row a cell b minus a end cell 0 row b cell a minus b end cell cell b minus a end cell row b 0 cell a minus b end cell end table close vertical bar equals 0$
$rightwards double arrow a open parentheses a minus b close parentheses to the power of 2 end exponent minus open parentheses b minus a close parentheses open square brackets b open parentheses a minus b close parentheses minus b open parentheses b minus a close parentheses close square brackets equals 0$
$rightwards double arrow a open parentheses a minus b close parentheses to the power of 2 end exponent minus 2 b blank open parentheses b minus a close parentheses open parentheses a minus b close parentheses equals 0$
$rightwards double arrow open parentheses a minus b close parentheses to the power of 2 end exponent open parentheses a minus 2 b close parentheses equals 0$
$rightwards double arrow a equals b$ or $a equals 2 b$
$rightwards double arrow fraction numerator a over denominator b end fraction equals 1$ or $fraction numerator a over denominator b end fraction equals 2$
$rightwards double arrow tan invisible function application x equals 1$ or $tan invisible function application x equals 2$
But we have $– pi divided by 4 less or equal than x less or equal than pi divided by 4$
$rightwards double arrow tan invisible function application left parenthesis pi divided by 4 right parenthesis less or equal than tan invisible function application x less or equal than tan invisible function application left parenthesis pi divided by 4 right parenthesis$
$rightwards double arrow negative 1 less or equal than tan invisible function application x less or equal than 1$
$therefore tan invisible function application x equals 1 blank rightwards double arrow x equals pi divided by 4$
Therefore, there is only one real root

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