Question

"Question48
Prove the identity : (1-cos² θ ) sec² θ = tan² θ
Chapter7,Trigonometric identities Exercise -7A , Page number 314"

Answer 1

To Prove : (1 -cos^2 Ф) sec^2 ФФ = tan^2 Ф

Proof : 

LHS

we know, (1 - cos^2 Ф ) = sin^2 Ф


So, 

( sin^2 Ф )( sec^2 Ф)

⇒ $\frac{height^2 }{hypotenuse^2}$ × $\frac{hypotenuse^2}{base^2}$



⇒ $\frac{height^2}{base^2}$

⇒ tan^2 Ф



Hence, Proved.

Answer 2

Solution :

1) To prove : “(1-cos² θ ) sec² θ = tan² θ ”
2) Inference ( Understanding the question, Thinking of the method ) : We shall start from Left hand side ( L. H. S) and proceed till we get R. H. S. We will use trigonometric identities too.

3) Formulas we are going to use :

$\rightarrow$ 1 -cos²θ = sin²θ
$\rightarrow$ tanθ = sinθ * secθ

4) Actual proof :

Firstly, We know that, sin²θ + cos²θ = 1 , So, 1 - cos²θ = sin²θ

$\\ \implies Left \: hand \: side \: \\ \implies ( 1 - cos^2 \theta) * ( sec^2 \theta) \\ \implies (sin^2 \theta) * sec^2 \theta \\ \implies ( sin^2 \theta) * \frac { tan^2 \theta } { sin^2 \theta } \\ \implies tan^2 \theta \\ \implies Right \: Hand \: side$

Therefore, We proved the identity (1-cos² θ ) sec² θ = tan² θ .

Hope it helps!