Question

prove with process: (1-cos²theta)(1+tan²theta)=tan²theta​

Answer 1

EXPLANATION.

⇒ (1 - cos²θ)(1 + tan²θ) = tan²θ.

As we know that,

Formula of :

⇒ sin²θ + cos²θ = 1.

⇒ 1 - cos²θ = sin²θ.

⇒ 1 + tan²θ = sec²θ.

Using this formula in the equation, we get.

⇒ (sin²θ)(sec²θ).

⇒ (sin²θ) x (1/cos²θ).

⇒ tan²θ.

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

$\huge\red{Answer:-}$

(1 - cos²θ) (1 + tan²θ) = tan²θ

L.H.S. = (1 - cos²θ) (1 + tan²θ)

= sin²θ × sec²θ

= sin²θ × 1/cos²θ

= sin²θ/cos²θ

= tan²θ = R.H.S.

$\fbox\blue{Hence, proved}$