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Maths-

General

Easy

Question

The number of solutions of $sec to the power of 2 end exponent invisible function application theta plus c o s e c to the power of 2 end exponent theta plus 2 blank c o s e c to the power of 2 end exponent theta equals 8 comma blank 0 less or equal than theta less or equal than pi divided by 2$ is

4
3
0
2

The correct answer is: 2

We have $blank fraction numerator 1 over denominator sin to the power of 2 end exponent invisible function application theta cos to the power of 2 end exponent invisible function application theta end fraction plus fraction numerator 2 over denominator sin to the power of 2 end exponent invisible function application theta end fraction 8 comma sin invisible function application theta not equal to 0 comma cos invisible function application theta not equal to 0$
$rightwards double arrow 1 plus 2 cos to the power of 2 end exponent invisible function application theta equals 8 sin to the power of 2 end exponent invisible function application theta cos to the power of 2 end exponent invisible function application theta equals 8 cos to the power of 2 end exponent invisible function application theta open parentheses 1 minus cos to the power of 2 end exponent invisible function application theta close parentheses$
$rightwards double arrow 8 cos to the power of 4 end exponent invisible function application theta minus 6 cos to the power of 2 end exponent invisible function application theta plus 1 equals 0$
$rightwards double arrow open parentheses 4 cos to the power of 2 end exponent invisible function application theta minus 1 close parentheses open parentheses 2 cos to the power of 2 end exponent invisible function application theta minus 1 close parentheses equals 0$
$rightwards double arrow cos to the power of 2 end exponent invisible function application theta equals 1 divided by 4 equals cos to the power of 2 end exponent invisible function application open parentheses pi divided by 3 close parentheses blank$ or $cos to the power of 2 end exponent invisible function application theta equals 1 divided by 2 equals cos to the power of 2 end exponent invisible function application open parentheses pi divided by 4 close parentheses$
$rightwards double arrow theta equals n pi plus-or-minus open parentheses pi divided by 3 close parentheses blank$ or $theta equals n pi plus-or-minus open parentheses pi divided by 4 close parentheses comma blank n element of Z$
Hence, for $0 less or equal than theta less or equal than pi divided by 2 comma blank theta equals pi divided by 3 comma blank theta equals pi divided by 4$

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