Question

If A and B are acute angle of right angled Triangle ABC the prove TanA.tanB is equal to 1

Answer 1

$\large\underline{\bf{Solution-}}$

Given :-

In triangle ABC,

• A and B are acute angles and triangle is right angled at C.

To Prove :-

• tanA × tanB = 1

Solution :-

Given that,

• Triangle ABC is right-angle triangle right-angled at C.

So,

⟼ Using angle sum property of triangle,

⟼ ∠A + ∠B + ∠C = 180°.

⟼ ∠A + ∠B = 180° - ∠C

⟼ ∠A + ∠B = 180° - 90°

⟼ ∠A + ∠B = 90°

⇛ ∠B = 90° - ∠A

Consider,

⟼ tanA × tanB

= tanA × tan(90° - A)

= tanA × cotA

$\: \: \: \: \: \: \: \: \: \: \: \: \: \red{\bigg \{ \because \: tan(90 \degree \: - x) = cotx\bigg \}}$

= 1

$\red{\bigg \{ \because \: tanx = \dfrac{1}{cotx}\rm :\implies\:tanx \times cotx = 1 \bigg \}}$

$\large{\boxed{\boxed{\bf{Hence, Proved}}}}$

Additional Information:-

Relationship between sides and T ratios

sin θ = Opposite Side/Hypotenuse

cos θ = Adjacent Side/Hypotenuse

tan θ = Opposite Side/Adjacent Side

sec θ = Hypotenuse/Adjacent Side

cosec θ = Hypotenuse/Opposite Side

cot θ = Adjacent Side/Opposite Side

Reciprocal Identities

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

sin θ = 1/cosec θ

cos θ = 1/sec θ

tan θ = 1/cot θ

Co-function Identities

sin (90°−x) = cos x

cos (90°−x) = sin x

tan (90°−x) = cot x

cot (90°−x) = tan x

sec (90°−x) = cosec x

cosec (90°−x) = sec x

Fundamental Trigonometric Identities

sin²θ + cos²θ = 1

sec²θ - tan²θ = 1

cosec²θ - cot²θ = 1