SOLUTION: Will you please help me with this equation: Given: sin (x+y)=sin x cos y + cos x sin y and cos (x+y) = cos x cos y - sin x sin y. Simplify ; sin ( x + pi ) × tan (x +pi/2) × sec

Algebra ->  Trigonometry-basics -> SOLUTION: Will you please help me with this equation: Given: sin (x+y)=sin x cos y + cos x sin y and cos (x+y) = cos x cos y - sin x sin y. Simplify ; sin ( x + pi ) × tan (x +pi/2) × sec      Log On


   

Question 1010663: Will you please help me with this equation: Given: sin (x+y)=sin x cos y + cos x sin y and cos (x+y) = cos x cos y - sin x sin y. Simplify ; sin ( x + pi ) × tan (x +pi/2) × sec (pi/2 - x)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let's break the problem into smaller pieces


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sin%28x%2Bpi%29+=+sin%28x%29%2Acos%28pi%29%2Bcos%28x%29%2Asin%28pi%29 Use the given identity


sin%28x%2Bpi%29+=+sin%28x%29%2A%28-1%29%2Bcos%28x%29%2A%280%29 Use the unit circle


sin%28x%2Bpi%29+=+-sin%28x%29%2B0 Multiply


sin%28x%2Bpi%29+=+-sin%28x%29 Simplify. We'll call this Equation A


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sin%28x%2Bpi%2F2%29+=+sin%28x%29%2Acos%28pi%2F2%29%2Bcos%28x%29%2Asin%28pi%2F2%29 Use the given identity


sin%28x%2Bpi%2F2%29+=+sin%28x%29%2A%280%29%2Bcos%28x%29%2A%281%29 Use the unit circle


sin%28x%2Bpi%2F2%29+=+0%2Bcos%28x%29 Multiply


sin%28x%2Bpi%2F2%29+=+cos%28x%29 Simplify. We'll call this Equation B


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cos%28x%2Bpi%2F2%29+=+cos%28x%29%2Acos%28pi%2F2%29-sin%28x%29sin%28pi%2F2%29 Use the given identity


cos%28x%2Bpi%2F2%29+=+cos%28x%29%2A%280%29-sin%28x%29%281%29 Use the unit circle


cos%28x%2Bpi%2F2%29+=+0-sin%28x%29 Multiply


cos%28x%2Bpi%2F2%29+=+-sin%28x%29 Simplify. We'll call this Equation C


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cos%28pi%2F2-x%29+=+cos%28pi%2F2%29%2Acos%28x%29%2Bsin%28pi%2F2%29sin%28x%29 Use the given identity


cos%28pi%2F2-x%29+=+0%2Acos%28x%29%2B1%2Asin%28x%29 Use the unit circle


cos%28pi%2F2-x%29+=+0%2Bsin%28x%29 Multiply


cos%28pi%2F2-x%29+=+sin%28x%29 Simplify. We'll call this Equation D


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sec%28pi%2F2-x%29+=+1%2F%28cos%28pi%2F2-x%29%29 Use the identity sec = 1/cos.


sec%28pi%2F2-x%29+=+1%2F%28sin%28x%29%29 Substitute (see Equation D). We'll call this Equation E


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tan%28x%2Bpi%2F2%29=%28sin%28x%2Bpi%2F2%29%29%2F%28cos%28x%2Bpi%2F2%29%29 Use the identity tan = sin/cos


tan%28x%2Bpi%2F2%29=%28cos%28x%29%29%2F%28cos%28x%2Bpi%2F2%29%29 Perform a substitution (see equation B)


tan%28x%2Bpi%2F2%29=%28cos%28x%29%29%2F%28-sin%28x%29%29 Perform a substitution (see equation C)


tan%28x%2Bpi%2F2%29=-%28cos%28x%29%29%2F%28sin%28x%29%29 Simplify. We'll call this Equation F


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Now onto the main problem


sin%28x%2Bpi%29%2Atan%28x%2Bpi%2F2%29%2Asec%28pi%2F2-x%29 Original Given Problem


-sin%28x%29%2Atan%28x%2Bpi%2F2%29%2Asec%28pi%2F2-x%29 Apply a substitution (see equation A)


-sin%28x%29%2A%28-%28cos%28x%29%29%2F%28sin%28x%29%29%29%2Asec%28pi%2F2-x%29 Apply a substitution (see equation F)


-sin%28x%29%2A%28-%28cos%28x%29%29%2F%28sin%28x%29%29%29%2A%281%2F%28sin%28x%29%29%29 Apply a substitution (see equation E)


%28-sin%28x%29%2A%28-cos%28x%29%29%2A1%29%2F%28sin%28x%29%2Asin%28x%29%29


%28sin%28x%29%2Acos%28x%29%29%2F%28sin%28x%29%2Asin%28x%29%29


%28highlight%28sin%28x%29%29%2Acos%28x%29%29%2F%28highlight%28sin%28x%29%29%2Asin%28x%29%29


%28cross%28sin%28x%29%29%2Acos%28x%29%29%2F%28cross%28sin%28x%29%29%2Asin%28x%29%29


%28cos%28x%29%29%2F%28sin%28x%29%29


cot%28x%29


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So, sin%28x%2Bpi%29%2Atan%28x%2Bpi%2F2%29%2Asec%28pi%2F2-x%29 simplifies to cot%28x%29