Advertisements
Advertisements
Question
Prove the following trigonometric identities.
(1 + cot A − cosec A) (1 + tan A + sec A) = 2
Advertisements
Solution
We have to prove (1 + cot A − cosec A) (1 + tan A + sec A) = 2
We know that, `sin^2 A + cos^2 A = 1`
So.
`(1 + cot A − cosec A) (1 + tan A + sec A) = (1 + cosA/sin A - 1/ sinA) (1 + sin A/cos A + 1/cos A)`
`= ((sin A + cos A - 1)/sin A)((cos A + sin A + 1)/cos A)`
`= ((sin A + cos A -1)(sin A + cos A + 1))/(sin A cos A)`
`= ({(sin A + cos A) - 1}{(sin A + cos A) + 1})/(sin A cos A)`
`= ((sin A + cos A)^2 -1)/(sin A cos A)`
`= (sin^2 A + 2 sin A cos A + cos^2 A - 1)/(sin A cos A)`
`= ((sin^2 A + cos^2 A) + 2 sin A cos A - 1)/(sin A cos A)`
`= (1 + 2 sin A cos A -1)/(sin A cos A)`
`= (2 sin A cos A)/(sin A cos A)`
= 2
Hence proved.
APPEARS IN
RELATED QUESTIONS
Prove the following identities, where the angles involved are acute angles for which the expressions are defined:
`(tan theta)/(1-cot theta) + (cot theta)/(1-tan theta) = 1+secthetacosectheta`
[Hint: Write the expression in terms of sinθ and cosθ]
Prove the following trigonometric identities
`cos theta/(1 - sin theta) = (1 + sin theta)/cos theta`
Prove the following trigonometric identities.
`(cos theta - sin theta + 1)/(cos theta + sin theta - 1) = cosec theta + cot theta`
`sin theta / ((1+costheta))+((1+costheta))/sin theta=2cosectheta`
Write the value of ` sec^2 theta ( 1+ sintheta )(1- sintheta).`
\[\frac{1 - \sin \theta}{\cos \theta}\] is equal to
The value of (1 + cot θ − cosec θ) (1 + tan θ + sec θ) is
Prove the following identity :
sinθcotθ + sinθcosecθ = 1 + cosθ
Prove the following identity :
`cosec^4A - cosec^2A = cot^4A + cot^2A`
Prove the following identity:
tan2A − sin2A = tan2A · sin2A
Prove the following identity :
`sinA/(1 + cosA) + (1 + cosA)/sinA = 2cosecA`
Prove the following identity :
`(1 + cotA + tanA)(sinA - cosA) = secA/(cosec^2A) - (cosecA)/sec^2A`
Without using trigonometric table , evaluate :
`cosec49°cos41° + (tan31°)/(cot59°)`
Prove that `(sin θ. cos (90° - θ) cos θ)/sin( 90° - θ) + (cos θ sin (90° - θ) sin θ)/(cos(90° - θ)) = 1`.
Without using a trigonometric table, prove that
`(cos 70°)/(sin 20°) + (cos 59°)/(sin 31°) - 8sin^2 30° = 0`.
Prove that `(tan θ + sin θ)/(tan θ - sin θ) = (sec θ + 1)/(sec θ - 1)`
Choose the correct alternative:
`(1 + cot^2"A")/(1 + tan^2"A")` = ?
Prove that cot2θ × sec2θ = cot2θ + 1
Prove that `(tan(90 - theta) + cot(90 - theta))/("cosec" theta)` = sec θ
If sinθ – cosθ = 0, then the value of (sin4θ + cos4θ) is ______.
