IMSEC in Collabora Office: Check Complex Secant Signs and Reciprocal Identities

• Introduced: the original Collabora release and launch date are not established by the reviewed sources.
• Availability: documented in Collabora Office 26.04 desktop help.
• Platform scope: online and mobile deployment support is not inferred from the desktop documentation.
• Checked: October 14, 2026. Function-specific documentation and independent controls; no native Collabora execution.

=IMSEC(Complex_number)

The documented forms use a+bi or a+bj text. Purely real input may also be numeric. Collabora describes the result as a string representing a complex number, with a zero part omitted from the display.

For example, its numeric control =IMSEC(2) returns approximately −2.40299796172238 as text. IMREAL can expose the corresponding real number explicitly; the displayed decimal does not by itself establish a numeric output type.

Secant is a reciprocal operation, not an inverse cosine that returns an angle.

Format A2:A5 as Text and enter 0.5+0.75j, 2+j, 0.5-0.75j and 2j. Calculate B with =IMSEC(A2) filled down. Use =IMREAL(B2) and =IMAGINARY(B2) in C and D.

The coefficients are independently calculated and rounded to nine decimals. Retain the complete B formulas for subsequent checks.

The first input’s cosine has a negative imaginary coefficient; taking its reciprocal changes that sign through conjugation. This is why the secant’s imaginary coefficient is positive in the first row.

For the first row, compare:

=IMDIV(COMPLEX(1;0;"j");IMCOS(A2))

Expected output coefficients match the table. The cosine denominator is approximately 1.136191474−0.394239642j, whose squared magnitude is approximately 1.446355961.

Its reciprocal divides the conjugate coefficients by that squared magnitude. It is not obtained by taking the reciprocal of each coefficient separately.

IMCOS and IMDIV make both stages inspectable. The explicit complex numerator avoids assuming an undocumented real-number coercion rule for the division function.

The official image uses denominator cosh(2b)+cos(2a), real numerator 2cos(a)cosh(b), and imaginary numerator 2sin(a)sinh(b).

For a=0.5 and b=0.75, check:

=2*COS(0.5)*COSH(0.75)/(COSH(1.5)+COS(1))
=2*SIN(0.5)*SINH(0.75)/(COSH(1.5)+COS(1))

The independently expected coefficients are 0.785554528 and 0.272574423, rounded. Both the plus in the denominator and the positive imaginary numerator matter.

The equation and the reciprocal construction describe the same function through different calculations. Agreement on this moderate fixture is useful evidence, not a universal stability guarantee at extreme inputs.

For A2, compare the complete secant square minus the complete tangent square:

=IMSUB(IMPRODUCT(IMSEC(A2);IMSEC(A2));IMPRODUCT(IMTAN(A2);IMTAN(A2)))

The independent target is real coefficient one and imaginary coefficient zero. The secant square has coefficients approximately 0.542799100 and 0.428244145; the tangent square has approximately −0.457200900 and 0.428244145.

The imaginary coefficients cancel on subtraction. IMTAN supplies a complete complex value; squaring only its real coefficient would not test the identity.

A simpler control multiplies IMSEC(A2) by IMCOS(A2), again giving mathematical quantity one. Use both extracted coordinates when checking either relation.

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

For purely imaginary inputs j and 2j, the expected real coefficients are approximately 0.648054274 and 0.265802229. These equal the reciprocals of cosh(1) and cosh(2). Their values below one do not violate a real-angle secant restriction, because the inputs are not real angles.

Secant is even: negating both coordinates of 0.5+0.75j leaves the result unchanged. Conjugating the input instead reverses its imaginary output sign, as in the third table row.

Cosine is zero at real pi/2 plus integer multiples of pi, giving mathematical secant poles. Native results or error codes at zero denominators and approximate PI()-based inputs are not established by this page.

The official 4−3i example gives coefficients approximately −0.0652940278579471 and 0.0752249603027732. The reference also names OpenDocument 1.2 membership; this does not date the first Collabora release.

Malformed text, mixed suffixes, blank coercion and extreme overflow behavior remain unverified. Keep numerical tolerances explicit rather than promising exact native output-string equality.

• Collabora Office 26.04: IMSEC.
• Editorial testing standards.

Complete dedicated reference and its equation image reviewed. Examples are independently calculated controls or explicit documented statements, not native application captures. Formulas use English names and semicolon separators.