Advertisements
Advertisements
प्रश्न
Prove the following trigonometric identities.
`(cos theta - sin theta + 1)/(cos theta + sin theta - 1) = cosec theta + cot theta`
उत्तर
We have to prove the following identity
`(cos theta - sin theta + 1)/(cos theta + sin theta - 1) = cosec theta + cot theta`
Consider the LHS = `(cos theta - sin theta + 1)/(cos theta + sin theta - 1)`
`= (cos theta - sin theta + 1)/(cos theta + sin theta - 1) xx (cos theta + sin theta + 1)/(cos theta + sin theta + 1)`
`= ((cos theta + 1)^2 - (sin theta)^2)/((cos theta + sin theta)^2 - (1)^2)`
`= (cos^2 theta + 1 + 2 cos theta - sin^2 theta)/(cos^2 theta + sin^2 theta + 2 cos theta sin theta - 1)`
`= (cos^2 theta + 1 + 2 cos theta - (1 - cos^2 theta))/(1 + 2 cos theta sin theta - 1)`
`= (2 cos^2 theta + 2 cos theta)/(2 cos theta sin theta)`
`= (2 cos^2 theta + 2 cos theta)/(2 cos theta sin theta)`
`= (2 cos theta(cos theta + 1))/(2 cos theta sin theta)`
`= (cos theta + 1)/sin theta`
`= cos theta/sin theta + 1/sin theta`
`= cot theta + cosec theta`
= RHS
APPEARS IN
संबंधित प्रश्न
If secθ + tanθ = p, show that `(p^{2}-1)/(p^{2}+1)=\sin \theta`
Prove the following trigonometric identities
(1 + cot2 A) sin2 A = 1
Prove that `(sec theta - 1)/(sec theta + 1) = ((sin theta)/(1 + cos theta))^2`
Prove the following identities:
`tan A - cot A = (1 - 2cos^2A)/(sin A cos A)`
Prove the following identities:
(cosec A – sin A) (sec A – cos A) (tan A + cot A) = 1
Prove the following identities:
`sinA/(1 + cosA) = cosec A - cot A`
Prove the following identities:
`(sinAtanA)/(1 - cosA) = 1 + secA`
If sec A + tan A = p, show that:
`sin A = (p^2 - 1)/(p^2 + 1)`
Write the value of cosec2 (90° − θ) − tan2 θ.
\[\frac{1 - \sin \theta}{\cos \theta}\] is equal to
Prove the following identity :
`(1 + tan^2A) + (1 + 1/tan^2A) = 1/(sin^2A - sin^4A)`
Prove that `(sin θ tan θ)/(1 - cos θ) = 1 + sec θ.`
Prove that cosec2 (90° - θ) + cot2 (90° - θ) = 1 + 2 tan2 θ.
Prove that `(sin 70°)/(cos 20°) + (cosec 20°)/(sec 70°) - 2 cos 70° xx cosec 20°` = 0.
Prove the following identities: cot θ - tan θ = `(2 cos^2 θ - 1)/(sin θ cos θ)`.
If tan θ = `9/40`, complete the activity to find the value of sec θ.
Activity:
sec2θ = 1 + `square` ......[Fundamental trigonometric identity]
sec2θ = 1 + `square^2`
sec2θ = 1 + `square`
sec θ = `square`
If tan θ = `7/24`, then to find value of cos θ complete the activity given below.
Activity:
sec2θ = 1 + `square` ......[Fundamental tri. identity]
sec2θ = 1 + `square^2`
sec2θ = 1 + `square/576`
sec2θ = `square/576`
sec θ = `square`
cos θ = `square` .......`[cos theta = 1/sectheta]`
Show that `(cos^2(45^circ + theta) + cos^2(45^circ - theta))/(tan(60^circ + theta) tan(30^circ - theta))` = 1
Let x1, x2, x3 be the solutions of `tan^-1((2x + 1)/(x + 1)) + tan^-1((2x - 1)/(x - 1))` = 2tan–1(x + 1) where x1 < x2 < x3 then 2x1 + x2 + x32 is equal to ______.
(1 – cos2 A) is equal to ______.
