Question

Plzzz help class 9 optional math

If tan theta =p show that sec theta*cosec theta =p+1/p​

Answer 1

`sec(\theta) * cosec(\theta) = (tan^2 (\theta)+ 1)/(tan (\theta)) = tan (\theta)+ (1)/(tan (\theta)) = p + (1)/(p)`

Explanation:

The given trigonometric relations are

tan(θ) = p

sec(θ)×cosec(θ) = p + 1/p

We note that, when tan(θ) = p, we have;

p + 1/p = tan(θ) + 1/(tan(θ)) = (tan²(θ) + 1)/tan(θ)

By trigonometric ratios, we have;

tan²(θ) + 1 = sec²(θ) =1/cos²(θ) which gives;

(tan²(θ) + 1)/tan(θ) = 1/cos²(θ) × 1/tan(θ) = cos(θ)/sin(θ)×1/cos²(θ)

`(1)/(cos^2(\theta)) * (cos (\theta))/(sin( \theta)) = (1)/(cos(\theta)) * (1)/(sin( \theta)) = sec(\theta) * cosec(\theta)`

Therefore;

`If \ tan (\theta) = p \ then \ sec(\theta) * cosec(\theta) = p + (1)/(p)`