TRIG_TR9

This transformation applies sum-to-product identities to rewrite additive trigonometric expressions as products.

A standard identity is:

\cos(a)+\cos(b)=2\cos\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right)

Product forms can reveal structure for factoring and further simplification.

Excel Usage

=TRIG_TR9(expression)

• expression (str, required): Expression with trig sums to convert (expression string).

Returns (str): Transformed expression as a string, or an error message.

Example 1: Converts cosine sum to product

Inputs:

expression
cos(1) + cos(2)

Excel formula:

=TRIG_TR9("cos(1) + cos(2)")

Expected output:

"2*cos(1/2)*cos(3/2)"

Example 2: Partially converts mixed sine and cosine sum

Inputs:

expression
cos(1) + 2sin(1) + 2sin(2)

Excel formula:

=TRIG_TR9("cos(1) + 2*sin(1) + 2*sin(2)")

Expected output:

"cos(1) + 4*sin(3/2)*cos(1/2)"

Example 3: Converts symbolic cosine sum

Inputs:

expression
cos(a) + cos(b)

Excel formula:

=TRIG_TR9("cos(a) + cos(b)")

Expected output:

"2*cos(a/2 - b/2)*cos(a/2 + b/2)"

Example 4: Converts sine sum to product

Inputs:

expression
sin(a) + sin(b)

Excel formula:

=TRIG_TR9("sin(a) + sin(b)")

Expected output:

"2*sin(a/2 + b/2)*cos(a/2 - b/2)"

Python Code

from sympy import sympify
from sympy.simplify.fu import TR9 as sympy_tr9

def trig_tr9(expression):
    """
    Convert sums of sine or cosine terms into product forms.

    See: https://docs.sympy.org/latest/modules/simplify/fu.html#sympy.simplify.fu.TR9

    This example function is provided as-is without any representation of accuracy.

    Args:
        expression (str): Expression with trig sums to convert (expression string).

    Returns:
        str: Transformed expression as a string, or an error message.
    """
    try:
        expr_text = expression.replace("^", "**") if isinstance(expression, str) else expression
        expr_obj = sympify(expr_text)
        result = sympy_tr9(expr_obj)
        return str(result)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

expression *

Expression with trig sums to convert (expression string).