Advertisements
Advertisements
प्रश्न
Prove the following trigonometric identities. `(1 - cos A)/(1 + cos A) = (cot A - cosec A)^2`
Advertisements
उत्तर १
We need to prove `(1 - cos A)/(1 + cos A) = (cot A - cosec A)^2`
Now, rationalising the L.H.S, we get
`(1 - cos A)/(1 + cos A) = ((1 - cos A)/(1 + cos A)) ((1 - cos A)/(1 - cos A))`
`= (1 - cos A)^2/(1 - cos^2 A)` (using `a^2 - b^2 = (a + b)(a - b))`
` = (1 + cos^2 A - 2 cos A)/sin^2 A` (Using `sin^2 theta = 1 - cos^2 theta`)
`= 1/sin^2 A + cos^2 A/sin^2 A - (2 cos A)/sin^2 A`
Using `cosec theta = 1/sin theta` and `cot theta = cos theta/sin theta` we get
`1/sin^2 A + cos^2 A/sin^2 A - (2 cos A)/sin^2 A = cosec^2 A + cot^2 A - 2 cot A cosec A`
` (cot A - cosec A)^2` (Using `(a + b)^2 = a^2 + b^2 + 2ab`)
Hence proved.
उत्तर २
LHS = `(1 - cos θ)/(1 + cos θ)`
= `(1 - cos θ)/(1 + cos θ) xx (1 - cos θ)/(1 - cos θ)`
= `(1 - cos θ)^2/(1 - cos^2 θ)`
= `(1 - cos θ)^2/(sin^2 θ)`
= `[(1 - cosθ)/(sin θ)]^2`
= `[ 1/sinθ - cosθ/sin θ ]^2`
= ( cosec θ - cot θ )2
= [ - (cot θ - cosec θ)]2
= (cot θ - cosec θ)2
= RHS
Hence proved.
APPEARS IN
संबंधित प्रश्न
Prove the following identities:
`(i) cos4^4 A – cos^2 A = sin^4 A – sin^2 A`
`(ii) cot^4 A – 1 = cosec^4 A – 2cosec^2 A`
`(iii) sin^6 A + cos^6 A = 1 – 3sin^2 A cos^2 A.`
If (secA + tanA)(secB + tanB)(secC + tanC) = (secA – tanA)(secB – tanB)(secC – tanC) prove that each of the side is equal to ±1. We have,
If secθ + tanθ = p, show that `(p^{2}-1)/(p^{2}+1)=\sin \theta`
Prove the following identities, where the angles involved are acute angles for which the expressions are defined:
`(cos A-sinA+1)/(cosA+sinA-1)=cosecA+cotA ` using the identity cosec2 A = 1 cot2 A.
Prove the following trigonometric identities.
`sqrt((1 - cos theta)/(1 + cos theta)) = cosec theta - cot theta`
Prove the following trigonometric identities.
`(1 - sin θ)/(1 + sin θ) = (sec θ - tan θ)^2`
Prove the following trigonometric identities.
`1/(1 + sin A) + 1/(1 - sin A) = 2sec^2 A`
Prove the following trigonometric identity.
`(sin theta - cos theta + 1)/(sin theta + cos theta - 1) = 1/(sec theta - tan theta)`
Prove the following trigonometric identities.
`1/(sec A + tan A) - 1/cos A = 1/cos A - 1/(sec A - tan A)`
Prove the following identities:
`(cotA - cosecA)^2 = (1 - cosA)/(1 + cosA)`
Prove the following identities:
`cosA/(1 + sinA) + tanA = secA`
If sec A + tan A = p, show that:
`sin A = (p^2 - 1)/(p^2 + 1)`
If 2 sin A – 1 = 0, show that: sin 3A = 3 sin A – 4 sin3 A
Show that none of the following is an identity:
`sin^2 theta + sin theta =2`
If` (sec theta + tan theta)= m and ( sec theta - tan theta ) = n ,` show that mn =1
If `sec theta + tan theta = x," find the value of " sec theta`
Write the value of cosec2 (90° − θ) − tan2 θ.
\[\frac{\sin \theta}{1 + \cos \theta}\]is equal to
2 (sin6 θ + cos6 θ) − 3 (sin4 θ + cos4 θ) is equal to
Prove the following identity :
cosecθ(1 + cosθ)(cosecθ - cotθ) = 1
Prove the following identity :
(secA - cosA)(secA + cosA) = `sin^2A + tan^2A`
Prove the following identity :
`(secθ - tanθ)^2 = (1 - sinθ)/(1 + sinθ)`
Prove the following identity :
`(1 + cotA)^2 + (1 - cotA)^2 = 2cosec^2A`
For ΔABC , prove that :
`sin((A + B)/2) = cos"C/2`
Prove that cot θ. tan (90° - θ) - sec (90° - θ). cosec θ + 1 = 0.
If cosθ + sinθ = `sqrt2` cosθ, show that cosθ - sinθ = `sqrt2` sinθ.
Prove that:
`(sin A + cos A)/(sin A - cos A) + (sin A - cos A)/(sin A + cos A) = 2/(2 sin^2 A - 1)`
The value of 2sinθ can be `a + 1/a`, where a is a positive number, and a ≠ 1.
sec θ when expressed in term of cot θ, is equal to ______.
