Question

1/1 + tan²A + 1/1 + cot²A = 1 (Prove it)

Answer 1

EXPLANATION.

⇒ 1/(1 + tan²A) + 1/(1 + cot²A) = 1.

As we know that,

From L.H.S.

⇒ 1/(1 + tan²A) + 1/(1 + cot²A).

As we know that,

Formula of :

⇒ 1 + tan²θ = sec²θ.

⇒ 1 + cot²θ = cosec²θ.

Using this formula in the equation, we get.

⇒ 1/(sec²A) + 1/(cosec²A).

⇒ cos²A + sin²A.

⇒ 1.

Hence proved.

                                                                                                                   

MORE INFORMATION.

Trigonometric ratios of multiple angles.

(1) = sin2θ = 2sinθ.cosθ = 2tanθ/1 + tan²θ.

(2) = cos2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ = 1 - tan²θ/1 + tan²θ.

(3) = tan2θ = 2tanθ/1 - tan²θ.

(4) = sin3θ = 3sinθ - 4sin³θ.

(5) = cos3θ = 4cos³θ - 3cosθ.

(6) = tan3θ = 3tanθ - tan³θ/1 - 3tan²θ.

Answer 2

Answer :-

We have :

$\rightarrow \bold{\dfrac{1}{1+tan^2A}+\dfrac{1}{1+cot^2A} }$

We know :

⇒ 1 + tan²A = sec²A

⇒ 1 + cot²A = cosec²A

Hence :

$\rightarrow \bold{\dfrac{1}{sec^2A}+\dfrac{1}{cosec^2A} }$

We know :

⇒ sec A = 1/cos A

⇒ cosec A = 1/sin A

Hence :

$\rightarrow \bold{\dfrac{1}{\dfrac{1}{cos^2A} }+\dfrac{1}{\dfrac{1}{sin2A} } }$

$\rightarrow \bold{cos^2A + sin^2A=1 }$

We know :

⇒ cos²A + sin²A = 1

∴ L.H.S = R.H.S, Hence proved !