Simplify

Problem

sin(π/5)*cos((3*π)/10)+cos(π/5)*sin((3*π)/10)

Solution

1. Identify the trigonometric identity that matches the structure of the expression. The expression follows the form sin(A)*cos(B)+cos(A)*sin(B)

2. Apply the formula for the sine of a sum, which states:

sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B)

3. Substitute the values A=π/5 and B=(3*π)/10 into the sum formula:

sin(π/5+(3*π)/10)

4. Find a common denominator to add the fractions inside the sine function:

π/5+(3*π)/10=(2*π)/10+(3*π)/10

5. Simplify the sum of the angles:

(2*π+3*π)/10=(5*π)/10

(5*π)/10=π/2

6. Evaluate the sine of the resulting angle:

sin(π/2)=1

Final Answer

sin(π/5)*cos((3*π)/10)+cos(π/5)*sin((3*π)/10)=1

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