Advertisements
Advertisements
Question
Prove the following trigonometric identities.
`(sec A - tan A)/(sec A + tan A) = (cos^2 A)/(1 + sin A)^2`
Solution
We need to prove `(sec A - tan A)/(sec A + tan A) = (cos^2 A)/(1 + sin A)^2`
Here, we will first solve the LHS.
Now using `sec theta = 1/cos theta` and `tan theta = sin theta/cos theta`, we get
`(sec A - tan A)/(sec A + tan A) = (1/cos A - sin A/cos A)/(1/cos A + sin A/cos A)`
`= ((1 - sin A)/cos A)/((1 + sin A)/cos A)`
`= (1 - sin A)/(1 + sin A)`
Further, multiplying both numerator and denominator by 1 + sin A we get
`(1 - sin A)/(1 + sin A) = ((1 - sin A)/(1 + sin A))((1 + sin A)/(1 = sin A))`
`= ((1 -sin A)(1 + sin A))/(1 + sin A)^2`
`= (1 s sin^2 A)/(1 + sin A)^2`
Now, using the property `cos^2 theta + sin^2 theta = 1`, we get
So,
`(1 - sin^2 A)/(1 + sin A)^2 = cos^2 A/(1 + sin A)^2` = RHS.
Hence proved
APPEARS IN
RELATED QUESTIONS
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,
Prove the following trigonometric identities
`cos theta/(1 - sin theta) = (1 + sin theta)/cos theta`
Prove the following trigonometric identities
tan2 A + cot2 A = sec2 A cosec2 A − 2
Prove the following trigonometric identities.
(sec A + tan A − 1) (sec A − tan A + 1) = 2 tan A
Prove the following identities:
`sqrt((1 - cosA)/(1 + cosA)) = cosec A - cot A`
Prove the following identities:
`(1+ sin A)/(cosec A - cot A) - (1 - sin A)/(cosec A + cot A) = 2(1 + cot A)`
Prove that:
2 sin2 A + cos4 A = 1 + sin4 A
Write the value of `(cot^2 theta - 1/(sin^2 theta))`.
If `cot theta = 1/ sqrt(3) , "write the value of" ((1- cos^2 theta))/((2 -sin^2 theta))`
Prove that:
(cosec θ - sinθ )(secθ - cosθ ) ( tanθ +cot θ) =1
Prove the following identity :
`1/(sinA + cosA) + 1/(sinA - cosA) = (2sinA)/(1 - 2cos^2A)`
Prove that:
`(cot A - 1)/(2 - sec^2 A) = cot A/(1 + tan A)`
Prove that sin2 θ + cos4 θ = cos2 θ + sin4 θ.
Prove that `(tan θ)/(cot(90° - θ)) + (sec (90° - θ) sin (90° - θ))/(cosθ. cosec θ) = 2`.
Prove that identity:
`(sec A - 1)/(sec A + 1) = (1 - cos A)/(1 + cos A)`
Prove that: `(sin A + cos A)/(sin A - cos A) + (sin A - cos A)/(sin A + cos A) = 2/(sin^2 A - cos^2 A)`.
Prove that `(1 + sintheta)/(1 - sin theta)` = (sec θ + tan θ)2
Prove that `sec"A"/(tan "A" + cot "A")` = sin A
Statement 1: sin2θ + cos2θ = 1
Statement 2: cosec2θ + cot2θ = 1
Which of the following is valid?
