Advertisements
Advertisements
Question
Prove the following identities.
`(cot theta - cos theta)/(cot theta + cos theta) = ("cosec" theta - 1)/("cosec" theta + 1)`
Advertisements
Solution
`(cot theta - cos theta)/(cot theta + cos theta) = ("cosec" theta - 1)/("cosec" theta + 1)`
L.H.S = `(cot theta - cos theta)/(cot theta + cos theta)`
= `cos theta/sin theta - cos theta ÷ cos theta/sin theta + cos theta`
= `(cos theta - sin theta cos theta)/sin theta ÷ (cos theta + sin theta cos theta)/sin theta`
= `(cos theta(1 - sin theta))/sin theta ÷ (cos theta(1 + sin theta))/sin theta`
= `(cos theta(1 - sin theta))/sin theta xx sin theta/(cos theta(1 + sin theta))`
= `(1 - sin theta)/(1 + sin theta)`
R.H.S = `("cosec" - 1)/("cosec"+1)`
= `1/sin theta - 1 ÷ 1/sin theta+ 1`
= `(1 - sin theta)/sin theta ÷ (1 + sin theta)/sin theta`
= `(1 - sin theta)/sin theta xx sin theta/(1 + sin theta)`
= `(1 - sin theta)/(1 + sin theta)`
R.H.S = L.H.S ⇒ `(cot theta - cos theta)/(cot theta + cos theta) = ("cosec" theta - 1)/("cosec" theta + 1)`
APPEARS IN
RELATED QUESTIONS
Prove the following trigonometric identities.
tan2 A sec2 B − sec2 A tan2 B = tan2 A − tan2 B
Prove the following identities:
`cosecA + cotA = 1/(cosecA - cotA)`
Prove that:
`tanA/(1 - cotA) + cotA/(1 - tanA) = secA "cosec" A + 1`
`sin^2 theta + cos^4 theta = cos^2 theta + sin^4 theta`
Write the value of `(1 + tan^2 theta ) cos^2 theta`.
Prove the following identity :
tanA+cotA=secAcosecA
Prove the following identity :
`(1 + sinA)/(1 - sinA) = (cosecA + 1)/(cosecA - 1)`
sin4A – cos4A = 1 – 2cos2A. For proof of this complete the activity given below.
Activity:
L.H.S. = `square`
= (sin2A + cos2A) `(square)`
= `1 (square)` ...`[sin^2"A" + square = 1]`
= `square` – cos2A ...[sin2A = 1 – cos2A]
= `square`
= R.H.S.
Prove that sin2A . tan A + cos2A . cot A + 2 sin A . cos A = tan A + cot A.
Prove that `(cot A + "cosec" A - 1)/(cot A - "cosec" A + 1) = (1 + cos A)/(sin A)`.
