Advertisements
Advertisements
Question
Prove the following trigonometric identities.
`cos A/(1 - tan A) + sin A/(1 - cot A) = sin A + cos A`
Advertisements
Solution
We need to prove `cos A/(1 - tan A) + sin A/(1 - cot A) = sin A + cos `
Solving the L.H.S, we get
`cos A/(1 - tan A) + sin A/(1 - cot A)`
= `cos A/(1 - sin A/cos A) + sin A/(1 - cos A/sin A)`
`= cos A/((cos A - sin A)/cos A) + sin A/((sin A - cos A)/sin A)`
`= cos^2 A/(cos A - sin A) + (sin^2 A)/(sin A - cos A)`
`= (cos^2 A - sin^2 A)/(cos A - sin A)`
`= ((cos A + sin A)(cos A - sin A))/(cos A - sin A)` [using `a^2 - b^2 = (a + b)(a -b)`]
= cos A + sin A
= RHS
Hence proved.
APPEARS IN
RELATED QUESTIONS
Prove the following trigonometric identity.
`cos^2 A + 1/(1 + cot^2 A) = 1`
Prove the following trigonometric identities.
sin2 A cot2 A + cos2 A tan2 A = 1
Prove the following trigonometric identities.
`tan theta/(1 - cot theta) + cot theta/(1 - tan theta) = 1 + tan theta + cot theta`
Prove the following trigonometric identities. `(1 - cos A)/(1 + cos A) = (cot A - cosec A)^2`
Prove the following trigonometric identities.
`(1 + cos theta + sin theta)/(1 + cos theta - sin theta) = (1 + sin theta)/cos theta`
Prove the following trigonometric identities
sec4 A(1 − sin4 A) − 2 tan2 A = 1
Prove the following identities:
`((1 + tan^2A)cotA)/(cosec^2A) = tan A`
Prove the following identities:
`(cotA + cosecA - 1)/(cotA - cosecA + 1) = (1 + cosA)/sinA`
Prove the following identities:
`((cosecA - cotA)^2 + 1)/(secA(cosecA - cotA)) = 2cotA`
Prove that:
cos A (1 + cot A) + sin A (1 + tan A) = sec A + cosec A
`sqrt((1+cos theta)/(1-cos theta)) + sqrt((1-cos theta )/(1+ cos theta )) = 2 cosec theta`
`(tan A + tanB )/(cot A + cot B) = tan A tan B`
What is the value of (1 + tan2 θ) (1 − sin θ) (1 + sin θ)?
\[\frac{\sin \theta}{1 + \cos \theta}\]is equal to
9 sec2 A − 9 tan2 A is equal to
Prove the following identity :
`(secA - 1)/(secA + 1) = (1 - cosA)/(1 + cosA)`
Prove the following identity :
`sqrt((1 + cosA)/(1 - cosA)) = cosecA + cotA`
Prove the following identity :
`(secθ - tanθ)^2 = (1 - sinθ)/(1 + sinθ)`
Prove the following identity :
`(cos^3A + sin^3A)/(cosA + sinA) + (cos^3A - sin^3A)/(cosA - sinA) = 2`
Prove the following identity :
`(cos^3θ + sin^3θ)/(cosθ + sinθ) + (cos^3θ - sin^3θ)/(cosθ - sinθ) = 2`
Prove that `sqrt(2 + tan^2 θ + cot^2 θ) = tan θ + cot θ`.
Prove that cosec2 (90° - θ) + cot2 (90° - θ) = 1 + 2 tan2 θ.
Prove that: `sqrt((1 - cos θ)/(1 + cos θ)) = cosec θ - cot θ`.
Without using trigonometric table, prove that
`cos^2 26° + cos 64° sin 26° + (tan 36°)/(cot 54°) = 2`
Prove that the following identities:
Sec A( 1 + sin A)( sec A - tan A) = 1.
Prove that `costheta/(1 + sintheta) = (1 - sintheta)/(costheta)`
Prove that `"cot A"/(1 - tan "A") + "tan A"/(1 - cot"A")` = 1 + tan A + cot A = sec A . cosec A + 1
Given that sin θ = `a/b`, then cos θ is equal to ______.
The value of 2sinθ can be `a + 1/a`, where a is a positive number, and a ≠ 1.
Prove that (sec θ + tan θ) (1 – sin θ) = cos θ
