Prove that (1 + sin θ)/(1 - sin θ) = (sec θ + tan θ)^2.

Prove that `(1 + sin θ)/(1 - sin θ) = (sec θ + tan θ)^2`.

सिद्धांत

L.H.S. = `(1 + sin θ)/(1 - sin θ)`

= `((1 + sinθ)/(cosθ))/((1 - sinθ)/(cosθ))` ...[Dividing numerator and denominator by cos θ]

= `(1/cosθ + (sinθ)/(cosθ))/(1/cosθ - (sinθ)/(cosθ)`

= `(secθ + tanθ)/(secθ - tanθ)`

= `(secθ + tanθ)/(secθ - tanθ) xx (secθ + tanθ)/(secθ + tanθ)` ...[On rationalising the denominator]

= `(secθ + tanθ)^2/(sec^2θ - tan^2θ)`

= `(secθ + tanθ)^2/1` ...`[(∵ 1 + tan^2θ = sec^2θ),(∴ sec^2θ - tan^2θ = 1)]`

= (sec θ + tan θ)²

= R.H.S.

∴ `(1 + sinθ)/(1 - sinθ) = (sec θ + tan θ)^2`

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पाठ 6: Trigonometry - Q.3 (B)

पाठ 6 Trigonometry
Q.3 (B) | Q 5.

Prove the following identities:

`( i)sin^{2}A/cos^{2}A+\cos^{2}A/sin^{2}A=\frac{1}{sin^{2}Acos^{2}A)-2`

`(ii)\frac{cosA}{1tanA}+\sin^{2}A/(sinAcosA)=\sin A\text{}+\cos A`

`( iii)((1+sin\theta )^{2}+(1sin\theta)^{2})/cos^{2}\theta =2( \frac{1+sin^{2}\theta}{1-sin^{2}\theta } )`

Prove the following trigonometric identities.

`(cos^2 theta)/sin theta - cosec theta + sin theta = 0`

Prove the following trigonometric identities.

`sin A/(sec A + tan A - 1) + cos A/(cosec A + cot A + 1) = 1`

if `a cos^3 theta + 3a cos theta sin^2 theta = m, a sin^3 theta + 3 a cos^2 theta sin theta = n`Prove that `(m + n)^(2/3) + (m - n)^(2/3)`

Prove the following identities:

(1 – tan A)² + (1 + tan A)² = 2 sec²A

Prove the following identities:

`secA/(secA + 1) + secA/(secA - 1) = 2cosec^2A`

If a cos `theta + b sin theta = m and a sin theta - b cos theta = n , "prove that "( m^2 + n^2 ) = ( a^2 + b^2 )`

Four alternative answers for the following question are given. Choose the correct alternative and write its alphabet:

sin θ × cosec θ = ______

(sec A + tan A) (1 − sin A) = ______.