The simplification of the left-hand side of the equation is x=2nπ+π/4 or 2nπ-π/4.
Given the equation is sin(x+π/4)-sin(x-π/4)=1.
Trigonometry is a branch of mathematics that deals with the study of the relationship between the sides of a triangle (right triangle) and its angles.
The given equation is sin(x+π/4)-sin(x-π/4)=1.
Here, we will apply the formulas
sin(A+B)=sinAcosB+cosAsinB and
sin(A-B)=sin AcosB-cosAsinB
Now, we will apply these formulas in left-hand side of equation.
Here, A=x and B=π/4.
L.H.S.=sin(x+π/4)-sin(x-π/4)
L.H.S.=sin x cos π/4+cos x sin π/4-(sin xcos π/4-cos x sin π/4)
L.H.S.=sin x cos π/4+cos x sin π/4-sin xcos π/4+cos x sin π/4
Now, we will cancel the terms, we get
L.H.S.=2cos x sin π/4
As we know, sin π/4=1/√2.
Now, substitute this value, we get
L.H.S=2cos x (1/√2)
Further, we will equate L.H.S. with R.H.S., we get
2cos x (1/√2)=1
Divide both sides with 2, we get
(2cos x (1/√2))/2=1/2
cos x (1/√2)=1/2
multiply both sides with √2, we get
√2 cos x(1/√2)=√2/2
cos x=√2/2
cos x=1/√2
x=cos⁻¹(1/√2)
x=2nπ+π/4 or 2nπ-π/4
Hence, the simplified form of the given equation sin(x+π/4)-sin(x-π/4)=1 is x=2nπ+π/4 or 2nπ-π/4.
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