# Cot-Tan formula

The law of cot or Tangent which is also called as a cot-tangent formula or cot-tangent rule is the ratio of the cot of the angle to the cos of the angle in tangent formula

Tan Theta = Opposite Side / Adjacent Side

Cot Theta = Adjacent Side/ Opposite Side

## Cot – Tan x formula

The TanÂ Î¸ is the ratio of the Opposite side to the Adjacent, where (Î¸) is one of the acute angles.

## Solved Examples

Out of other 6 trigonometric formulas, letâ€™s have a look at the practice question of tan theta formula.

Example 1: Show that (cosec A â€“ sin A)(sec A â€“ cos A) = 1/ (tan A + cot A)

Solution:
LHS = (cosec A â€“ sin A)(sec A â€“ cos A)
= [(1/sin A) – sin A][(1/cos A) – cos A] [(1 – sin2A)/sin A][(1 – cos2A)/cos A] = (cos2A/sin A)(sin2A/cos A)
= sin2A cos2A/sin A cos A
= sin A cos A

RHS = 1/(tan A + cot A)
= 1/[tan A + (1/tan A)] = tan A/ (tan2A + 1)
= tan A/sec2A
= tan A cos2A
= (sin A/cos A) cos2A
= sin A cos A
Therefore, LHS = RHS
(cosec A â€“ sin A)(sec A â€“ cos A) = 1/ (tan A + cot A)

Example 2: If âˆš3 tan Î¸ = 1, then find the value of cot2Î¸ + sin2Î¸ â€“ cos2Î¸.

Solution:
Given,
âˆš3 tan Î¸ = 1
tan Î¸ = 1/âˆš3
cot Î¸ = âˆš3

AC = âˆš(3 + 1) = âˆš4 = 2
sin Î¸ = Â½
cos Î¸ = âˆš3/2
cot2Î¸ + sin2Î¸ â€“ cos2Î¸ = (âˆš3)2 + (Â½)2 – (âˆš3/2)2
= 3 + Â¼ – Â¾
= (12 + 1 – 3)/4
= 5/2

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