Definitions [2]
Complementary angles: When the sum of the measures of two angles is 90°, the angles are called complementary angles. Example, 30° + 60° = 90°.
Complementary angles: When the sum of the measures of two angles is 90°, the angles are called complementary angles. Example, 30° + 60° = 90°.
Formulae [1]
For an acute angle A,
- sin (90° - A) = cos A
- cos (90° - A) = sin A
- tan (90° - A) = cot A
- cot (90° - A) = tan A
- sec (90° - A) = cosec A
- cosec (90° - A) = sec A
Theorems and Laws [2]
If tan A = cot B, prove that A + B = 90°.
∵ tan A = cot B
tan A = tan (90° – B)
A = 90° – B
A + B = 90°. Proved
Prove that `(tan A)/(cot A) = (sec^2A)/("cosec"^2A)`.
R.H.S. = `(sec^2A)/("cosec"^2A)`
= `(1 + tan^2A)/(1 + cot^2A)` ...`[(∵ 1 + tan^2A = sec^2A),(1 + cot^2A = "cosec"^2A)]`
= `(1 + (sin^2A)/(cos^2A))/(1 + (cos^2A)/(sin^2A))`
= `((cos^2A + sin^2A)/(cos^2A))/((sin^2A + cos^2A)/(sin^2A))`
= `(1/(cos^2A))/(1/(sin^2A))` ...[∵ sin2A + cos2A = 1]
= `(sin^2A)/(cos^2A)`
= tan2A
= tan A . tan A
= `(tan A)/(cot A)`
= L.H.S.
∴ `(tan A)/(cot A) = (sec^2A)/("cosec"^2A)`
