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

  1. a b less or equal than 0
  2. a b greater or equal than 0
  3. a plus b equals 0
  4. a b equals0

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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