Question

prove the folowing relation tan(45degree + A) .tan (45degree-A ) =1 give answer me step by step

Answer 1

Step-by-step explanation:

Given:-

Tan (45°+A) Tan (45°-A)

To find:-

Tan (45°+A) Tan (45°-1) = 1

Solution:-

Method-1:-

Given that

Tan (45°+A) Tan (45°-A)

Tan (45°+A) Tan (90-(45°+A)

We know that

Tan (90°-A) = Cot A

=> Tan (45°+A) Cot (45°+A)

We know that

Cot A = 1 / Tan A

=> Tan (45°+A) / Tan (45°+A)

=> 1

Method-2:-

Given that

Tan (45°+A) Tan (45°-A)

We know that

Tan (A+B) = (Tan A + Tan B )/ (1-Tan A Tan B)

And

Tan(A-B) = (Tan A- Tan B )/(1+Tan A Tan B)

We have A = 45° and B = A

=>[(Tan45° + Tan A )/ (1-Tan45° TanA)]×

[(Tan45°-Tan A)/(1+Tan 45° Tan A)]

We know that Tan 45°=1

=>[ (1+TanA)/(1-TanA)]×[(1-TanA)/(1+TanA)]

=> (1+TanA)(1-TanA)/(1-TanA)(1+TanA)

=>1

Method-3:-

Given that

Tan (45°+A) Tan (45°-A)

Tan (90°-(45°-A) Tan(45°-A)

We know that

Tan (90°-A) = Cot A

=> Tan (45°-A) Cot (45°-A)

We know that

Cot A = 1 / Tan A

=> Tan (45°-A) / Tan (45°-A)

=> 1

Answer:-

Tan (45°+A) Tan (45°-1) = 1

Used formulae:-

Tan (90°-A) = Cot A

Tan (A+B) = (Tan A + Tan B )/ (1-Tan A Tan B)

Tan(A-B) = (Tan A- Tan B )/(1+Tan A Tan B)