IMSEC in Excel: Check Complex Secant with Reciprocal and Square Identities

• Introduced: the Microsoft function catalogue marks this identity as an Excel 2013 addition. An exact first build or preview date is not established.
• Availability: official help lists Excel for Microsoft 365, Excel for Microsoft 365 for Mac, Excel 2024, Excel 2024 for Mac, Excel 2021, Excel 2021 for Mac, Excel 2019, Excel 2016.
• Checked: October 14, 2026; official IMSEC reference. Documentation and independent models; no native Excel execution or wider platform certification.

=IMSEC(inumber)

Inumber is required and uses x+yi or x+yj text. Microsoft recommends COMPLEX for converting numeric real and imaginary coefficients into that representation.

Secant is not inverse cosine. The former takes a reciprocal; the latter recovers an angle under a chosen inverse convention. These operations require different formulas and are not substitutes for one another.

All worked controls here construct complex arguments explicitly, including real-axis inputs. The dedicated reference is not a complete specification of plain-number, blank or mixed-suffix coercion.

Enter only the numeric input columns in A2:B5. In C2 use =COMPLEX(A2,B2); in D2 use =IMSEC(C2). Fill down. E and F can extract the numeric coefficients with IMREAL and IMAGINARY.

The output columns are rounded independent expectations, not input data. Retain D’s calculated value when comparing equivalent formulas.

The final value is positive and smaller than one. Real-angle secant restrictions do not prohibit it, because its argument is purely imaginary rather than a real angle.

For each row, compare the result with:

=IMDIV(COMPLEX(1,0),IMCOS(C2))

IMCOS supplies the full denominator, and IMDIV calculates the complete complex reciprocal.

For the first row, cosine has squared magnitude approximately 0.563466899. If cosine has coefficients r and s, the reciprocal uses r/(r²+s²) and −s/(r²+s²).

This shared denominator explains why dividing only real parts is insufficient. The first row’s cosine has a negative imaginary coefficient, while its reciprocal has a positive one through conjugation.

A multiply-back control, =IMPRODUCT(D2,IMCOS(C2)), has mathematical target one with zero imaginary coefficient. Fill down and test both extracted components.

An independently derived model for secant of a+bi uses denominator COSH(2b)+COS(2a). Its real numerator is 2COS(a)COSH(b), and its imaginary numerator is 2SIN(a)SINH(b).

For the first input, use:

=2*COS(1)*COSH(0.5)/(COSH(1)+COS(2))
=2*SIN(1)*SINH(0.5)/(COSH(1)+COS(2))

The independent targets are 1.081268323 and 0.778194035. The plus in the denominator is important; switching it to minus would correspond to a different reciprocal construction.

This component derivation is a mathematical check, not an equation printed in the dedicated Microsoft article. Use moderate inputs and explicit tolerances rather than assuming every rearrangement has identical floating-point behavior.

For the first row, compare:

=IMSUB(IMPRODUCT(IMSEC(C2),IMSEC(C2)),IMPRODUCT(IMTAN(C2),IMTAN(C2)))

The independent target is one, with zero imaginary coefficient. The secant square has coefficients approximately 0.563555230 and 1.682873119. The tangent square has approximately −0.436444770 and 1.682873119.

Subtracting cancels the imaginary terms and leaves one. IMTAN must be squared as a complete complex quantity using IMPRODUCT. Neither the square of its real coefficient nor the square of its magnitude is the same operation.

Check component differences against an explicit tolerance such as 1E-9 for this fixture. Printed coefficients have been rounded, so do not expect identities evaluated from the rounded table to be exact.

For a real sixty-degree angle, =IMSEC(COMPLEX(RADIANS(60),0)) has mathematical target two with imaginary coefficient zero.

For imaginary inputs i and 2i, the independent real outputs are approximately 0.648054274 and 0.265802229, with imaginary coefficient zero. These are reciprocals of COSH(1) and COSH(2).

Complete input negation leaves secant unchanged. For the first row, test =IMSEC(COMPLEX(-1,-0.5)). Conjugating the first input instead changes only the sign of the output’s imaginary coefficient.

These checks distinguish even symmetry from conjugation and expose a common sign error in manually assembled reciprocals.

Microsoft specifies #NUM! for unsupported complex-text format and #VALUE! for logical input. Its =IMSEC("4+3i") example has coefficients approximately −0.0652940278579471 and −0.0752249603027732, consistent with independent division.

Cosine has mathematical zeros at real pi/2 plus integer multiples of pi. The dedicated page does not establish native errors or exact outcomes at floating approximations to those points. Keep these pole cases separate from its explicit malformed-input errors.

Extreme overflow, underflow, mixed suffixes and a full coercion table remain unverified. Numerical component checks are preferable to promises of exact output-string equality.

• Microsoft IMSEC reference.
• Editorial testing standards.

Expected outputs are independently calculated examples or documented rules, not captured native Excel results. Formulas use English function names and comma argument separators. Locale, rendering and boundary restrictions are stated where relevant.