If f(x) = sinx +sin? (x + 7/3) + cos2.cos (x + (x+5) and g(5/4) = 1, then (gof)(x) =

Asked by adithyavardhansudagoni2004 | 27th Sep, 2020, 11:08: AM

Expert Answer:

The question must be:
If f(x) = sin2x + sin2(x+pi/3) + cosx cos(x+pi/3) and g(5/4) = 1, then gof(x) = ?
Solution:
Now, let us simplify f(x)
straight f open parentheses straight x close parentheses equals sin squared straight x plus open parentheses sinx space cos straight pi over 3 space plus thin space cosx space sin straight pi over 3 close parentheses squared plus cosx open parentheses cosx space cos straight pi over 3 minus sinx space sin straight pi over 3 close parentheses
equals sin squared straight x plus open parentheses sinx over 2 plus fraction numerator square root of 3 over denominator 2 end fraction cosx close parentheses squared plus cosx open parentheses cosx over 2 minus fraction numerator square root of 3 over denominator 2 end fraction sinx close parentheses
equals sin squared straight x plus fraction numerator sin squared straight x over denominator 4 end fraction plus 3 over 4 cos squared straight x plus fraction numerator square root of 3 over denominator 2 end fraction sinx space cosx plus fraction numerator cos squared straight x over denominator 2 end fraction minus fraction numerator square root of 3 over denominator 2 end fraction sinx space cosx
equals 5 over 4 sin squared straight x plus 5 over 4 cos squared straight x
equals 5 over 4
rightwards double arrow straight f left parenthesis straight x right parenthesis equals 5 over 4
Now comma space gof left parenthesis straight x right parenthesis space equals space straight g left parenthesis straight f left parenthesis straight x right parenthesis right parenthesis space equals space straight g open parentheses 5 over 4 close parentheses equals 1
Hence comma space gof left parenthesis straight x right parenthesis equals 1

Answered by Renu Varma | 27th Sep, 2020, 12:37: PM

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