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Maths-

General

Easy

Question

If $fraction numerator sec to the power of 8 space theta over denominator a end fraction plus fraction numerator tan to the power of 8 begin display style space end style theta over denominator b end fraction equals fraction numerator 1 over denominator a plus b end fraction$ then

$a b \leq 0$
$a b \geq 0$
$a + b = 0$
$a b =$ 0

hint Hint:

In the question we will simplify the LHS using trigonometric identities.

The correct answer is: $a b less or equal than 0$

$fraction numerator sec to the power of 8 space theta over denominator a end fraction plus fraction numerator tan to the power of 8 begin display style space end style theta over denominator b end fraction equals fraction numerator 1 over denominator a plus b end fraction rightwards double arrow fraction numerator begin display style fraction numerator 1 over denominator cos to the power of 8 theta end fraction end style over denominator a end fraction plus fraction numerator begin display style fraction numerator sin to the power of 8 theta over denominator cos to the power of 8 theta end fraction end style over denominator b end fraction equals fraction numerator 1 over denominator a plus b end fraction rightwards double arrow fraction numerator b plus a sin to the power of 8 theta over denominator a b cos to the power of 8 theta end fraction equals fraction numerator 1 over denominator a plus b end fraction rightwards double arrow a b plus a squared sin to the power of 8 theta plus b squared plus a b sin to the power of 8 theta equals a b cos to the power of 8 theta rightwards double arrow a b plus a squared sin to the power of 8 theta plus b squared plus a b sin to the power of 8 theta equals a b open parentheses 1 minus sin squared theta close parentheses to the power of 4 rightwards double arrow a b plus a squared sin to the power of 8 theta plus b squared plus a b sin to the power of 8 theta equals a b open parentheses 1 minus 4 sin squared theta plus 6 sin to the power of 4 theta minus 4 sin to the power of 6 theta plus sin to the power of 8 theta close parentheses space space space space space space space space open square brackets open parentheses a minus b close parentheses to the power of 4 equals a squared minus 4 a b squared plus 6 a squared b squared minus 4 a cubed b plus b to the power of 4 close square brackets rightwards double arrow a b plus a squared sin to the power of 8 theta plus b squared plus a b sin to the power of 8 theta equals a b minus 4 a b sin squared theta plus 6 a b sin to the power of 4 theta minus 4 a b sin to the power of 6 theta plus a b sin to the power of 8 theta rightwards double arrow a squared sin to the power of 8 theta plus b squared equals 6 a b sin to the power of 4 theta minus 4 a b sin to the power of 6 theta minus 4 a b sin squared theta rightwards double arrow open parentheses a sin to the power of 4 theta close parentheses squared plus b squared minus 2 a b sin to the power of 4 theta equals 4 a b sin squared theta open parentheses sin squared theta minus 1 minus sin to the power of 4 theta close parentheses rightwards double arrow open parentheses a sin to the power of 4 theta plus b close parentheses squared equals 4 a b sin squared theta open parentheses negative cos squared theta minus sin to the power of 4 theta close parentheses rightwards double arrow open parentheses a sin to the power of 4 theta plus b close parentheses squared equals negative 4 a b sin squared theta open parentheses cos squared theta plus sin to the power of 4 theta close parentheses rightwards double arrow a comma b space a r e space o f space o p p o s i t e space s i g n space rightwards double arrow a b less or equal than 0$

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