Advertisements
Advertisements
प्रश्न
If `sec theta + tan theta = p,` prove that
(i)`sec theta = 1/2 ( p+1/p) (ii) tan theta = 1/2 ( p- 1/p) (iii) sin theta = (p^2 -1)/(p^2+1)`
Advertisements
उत्तर
(i) We have , `sec theta + tan theta = p` ....................(1)
`⇒ (sec theta + tan theta )/1 xx (sec theta - tan theta )/( sec theta - tan theta ) = p`
`⇒ (sec ^2 theta - tan^2 theta )/( sec theta - tan theta) = p`
`⇒ 1/ (sec theta - tan theta ) =p`
`⇒ sec theta - tan theta = 1/ p` .........................(2)
Adding (1) and (2) , We get
2` sec theta = p + 1/p`
`⇒ sec theta = 1/2 ( p+1/p)`
(ii) subtracting (2) feom (1) , We get
`2 tan theta = (p - 1/p)`
`⇒ tan theta = 1/2 ( p-1/p)`
(iii) Using (i) and (ii) , We get
`sin theta = tantheta/ sec theta`
=`(1/2(p-1/p))/(1/2 (p+1/p)`
=`(((p^2-1)/p))/(((p^2+1))/p)`
∴ `sin theta = (p^2-1)/(p^2 +1)`
APPEARS IN
संबंधित प्रश्न
Prove the following trigonometric identities.
`(1 - sin theta)/(1 + sin theta) = (sec theta - tan theta)^2`
Prove the following trigonometric identity:
`sqrt((1 + sin A)/(1 - sin A)) = sec A + tan A`
Prove the following identities:
sec2 A . cosec2 A = tan2 A + cot2 A + 2
If tan A = n tan B and sin A = m sin B, prove that:
`cos^2A = (m^2 - 1)/(n^2 - 1)`
Prove that:
(cosec A – sin A) (sec A – cos A) sec2 A = tan A
`1+(tan^2 theta)/((1+ sec theta))= sec theta`
If`( 2 sin theta + 3 cos theta) =2 , " prove that " (3 sin theta - 2 cos theta) = +- 3.`
Prove that:
`"tanθ"/("secθ" – 1) = (tanθ + secθ + 1)/(tanθ + secθ - 1)`
Write the value of \[\cot^2 \theta - \frac{1}{\sin^2 \theta}\]
Prove the following identity :
`1/(tanA + cotA) = sinAcosA`
Without using trigonometric identity , show that :
`sec70^circ sin20^circ - cos20^circ cosec70^circ = 0`
Prove that:
`sqrt((sectheta - 1)/(sec theta + 1)) + sqrt((sectheta + 1)/(sectheta - 1)) = 2cosectheta`
Prove that `(sin θ tan θ)/(1 - cos θ) = 1 + sec θ.`
Prove that sin θ sin( 90° - θ) - cos θ cos( 90° - θ) = 0
Prove the following identities.
tan4 θ + tan2 θ = sec4 θ – sec2 θ
Prove the following identities.
`costheta/(1 + sintheta)` = sec θ – tan θ
If `sqrt(3)` sin θ – cos θ = θ, then show that tan 3θ = `(3tan theta - tan^3 theta)/(1 - 3 tan^2 theta)`
Choose the correct alternative:
`(1 + cot^2"A")/(1 + tan^2"A")` = ?
Prove that `sqrt((1 + cos "A")/(1 - cos"A"))` = cosec A + cot A
Show that tan 7° × tan 23° × tan 60° × tan 67° × tan 83° = `sqrt(3)`
