I can not seem to solve the following question: MacGyver has found himself in on

Question

I can not seem to solve the following question:

MacGyver has found himself in one of his usual life-threatening situations. He is in a closed room that has a mechanical device in the middle that is used to open the only entrance. Through his uncanny abilities, he determines that to get out of room, he will need to enter an expression into the machine that is equivalent to: 2cos(4g)[(square root 1 - cos^2(7g))cos(3g) - cos(7g) . (1/csc(3 g))] + g. with g being an angle in the first quadrant. Unfortunately, the machine will accept only ten characters, and cos(g) alone is six characters, counting the parenthesis, which this machine requires. If he enters an incorrect formula, the whole room will explode, so accuracy is important.

Explanation / Answer

we have to simplify the given expression

2cos(4g) [ (sqrt(1 -cos^2 7g)) cos(3g) -cos(7g) . 1/csc(3g) ] + g

now normally sin^2 x + cos^2 x =1

sin^2 x = 1 -cos^2 x

so Sinx= sqrt(1 - Cos^2 x)

and 1/csc3g = sin3g , So plug theese values in the given equation

so the equation becomes

2cos(4g) [ (sqrt(1 -cos^2 7g)) cos(3g) -cos(7g) . 1/csc(3g) ] + g = 2cos(4g) [ sin(7g) cos(3g) - cos(7g) sin(3g) ] +g

now sin(A-B) = SinAcosB - CosAsinB

here sin(7g) cos(3g) - cos(7g) sin(3g) we can reduce this

A=7g B=3g

sin(7g) cos(3g) - cos(7g) sin(3g = Sin(7g -3g) = sin(4g)

2cos(4g) [ (sqrt(1 -cos^2 7g)) cos(3g) -cos(7g) . 1/csc(3g) ] + g = 2cos(4g) sin(4g) + g

now sin2x = 2sinxcosx formula

so 2cos(4g) sin(4g) = sin(2x4g) [ using sin2x formula where x =4g]

= Sin(8g)

2cos(4g) [ (sqrt(1 -cos^2 7g)) cos(3g) -cos(7g) . 1/csc(3g) ] + g = sin(2*4g) +g

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