Question

prove that underoot(sec^2 theta(1-cos^2 theta)=tan theta​

Answer 1

Given to prove

$\sf\sqrt{sec^2\theta \times (1-cos^2\theta)}=tan\theta$

Formula to know:-

We know that

sin²A + cos²A = 1

sin²A = 1-cos²A

tanA = sinA/cosA

secA = 1/cosA

a²b² = (ab)²

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$\sf\sqrt{sec^2\theta \times (1-cos^2\theta)}=tan\theta$

Take LHS

$\sf\sqrt{sec^2\theta \times (1-cos^2\theta)}$

$\sf{1-cos^2A = sin^2A}$

$\sf\sqrt{sec^2\theta \times sin^2\theta}$

$\sf{a^2b^2 = (ab)^2}$

$\sf\sqrt{(sec\theta sin\theta)^2}$

secθ sinθ

$\sf\dfrac{1}{cos\theta} = sec\theta$

$\sf\dfrac{1}{cos\theta}$ sinθ

$\sf\dfrac{sin\theta}{cos\theta}$

tanθ

LHS = RHS

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know more:-

Trigon metric Identities

sin²θ + cos²θ = 1

sec²θ - tan²θ = 1

csc²θ - cot²θ = 1

Trigometric relations

sinθ = 1/cscθ

cosθ = 1 /secθ

tanθ = 1/cotθ

tanθ = sinθ/cosθ

cotθ = cosθ/sinθ

Trigonmetric ratios

sinθ = opp/hyp

cosθ = adj/hyp

tanθ = opp/adj

cotθ = adj/opp

cscθ = hyp/opp

secθ = hyp/adj