Simplify the expression (sin(a)cos(b)+cos(a)sin(b))/(cos(a)cos(b)-sin(a)sin(b)) * (cos(a)cos(b)+sin(a)sin(b))/(sin(a)cos(b)-cos(a)sin(b))sin(a)cos(b)+cos(a)sin(b)cos(a)cos(b)−sin(a)sin(b)⋅cos(a)cos(b)+sin(a)sin(b)sin(a)cos(b)−cos(a)sin(b)?
2 Answers
tan(a+b) * cot(a-b) tan(a+b)⋅cot(a−b)
Explanation:
The expression is:
E=(sin(a)cos(b)+cos(a)sin(b))/(cos(a)cos(b)-sin(a)sin(b)) * (cos(a)cos(b)+sin(a)sin(b))/(sin(a)cos(b)-cos(a)sin(b)) E=sin(a)cos(b)+cos(a)sin(b)cos(a)cos(b)−sin(a)sin(b)⋅cos(a)cos(b)+sin(a)sin(b)sin(a)cos(b)−cos(a)sin(b)
We can us the sine and cosine sum identities:
sin(A+B)=sinAcosB+cosAsinB sin(A+B)=sinAcosB+cosAsinB
sin(A-B)=sinAcosB-cosAsinB sin(A−B)=sinAcosB−cosAsinB
cos(A+B)=cosAcosB-sinAsinB cos(A+B)=cosAcosB−sinAsinB
cos(A-B)=cosAcosB+sinAsinB cos(A−B)=cosAcosB+sinAsinB
Applying these identities we can rewrite the expression as:
E = (sin(a+b))/(cos(a+b)) * (cos(a-b))/(sin(a-b)) E=sin(a+b)cos(a+b)⋅cos(a−b)sin(a−b)
\ \ \ = tan(a+b) * cot(a-b)
Explanation:
To solve this, you need to know these formulae:
If you replace
Similarly, replacing
Now let's move on to the question.
Simplifying all the terms, we get