Question

tan A + cot A = sec A cosec A​

Answer 1

Step-by-step explanation:

LHS =

tan A +cot A = sec A cosec A

sinA/cos A + cos A / sin A

sin² A + cos² A / cos A sin A

1 / cosA sinA ( sin²A + cos² A = 1)

sec A cosec A

LHS = RHS

Proved

Answer 2

$\maltese$Given to prove:-

$tanA+cotA = secA\: cosecA$

$\maltese$Solution :-

Take L.H.S tanA+cotA

From trigonometric relations ,

$tanA =\dfrac{sinA}{cosA}$

$cotA = \dfrac{cosA}{sinA}$

Substituting the values

$tanA +cotA = \dfrac{sinA}{cosA } +\dfrac{cosA}{sinA}$

Take L.C.M to the denominator

$= \dfrac{sinA(sinA)+cosA(cosA)}{sinA\:cosA}$

$= \dfrac{sin^2A+cos^2A}{sinA\:cosA}$

 From trigonometric identities,

sin^2A +cos^2A = 1

$=\dfrac{1}{sinA\:cosA}$

$= \dfrac{1}{sinA } \times \dfrac{1}{cosA}$

From trigonometric relations , We know that

1/sinA = cosecA and 1/cosA = secA

$= cosecA\times secA$

$= secA \:cosecA$

Hence, L.H.S = R.H.S

Proved !

$\maltese$Know more:-

$\maltese$ Trigonometric Identities

sin²θ + cos²θ = 1

sec²θ - tan²θ = 1

csc²θ - cot²θ = 1

$\maltese$Trigonometric relations

sinθ = 1/cscθ

cosθ = 1 /secθ

tanθ = 1/cotθ

tanθ = sinθ/cosθ

cotθ = cosθ/sinθ

$\maltese$Trigonometric ratios

sinθ = opp/hyp

cosθ = adj/hyp

tanθ = opp/adj

cotθ = adj/opp

cscθ = hyp/opp

secθ = hyp/adj