Advertisements
Advertisements
प्रश्न
Prove the following trigonometric identities.
`(tan^2 A)/(1 + tan^2 A) + (cot^2 A)/(1 + cot^2 A) = 1`
Advertisements
उत्तर
In the given question, we need to prove `(tan^2 A)/(1 + tan^2 A) + (cot^2 A)/(1 + cot^2 A) = 1`
Here, we will first solve the LHS.
Now using `tan theta = sin theta/cos theta` and `cot theta = cos theta/sin theta` we get
`tan^2 A/(1 + tan^2 A) + cot^2 A/(1 + cot^2 A) = ((sin^2 A/cos^2 A))/((1 + sin^2 A/cos^2 A)) + ((cos^2 A/sin^2 A))/((1 + cos^2 A/sin^2 A))`
`= ((sin^2 A/cos^2 A))/(((cos^2 + sin^2 A)/cos^2 A)) + ((cos^2 A/sin^2 A))/(((sin^2 A + cos^2 A)/sin^2 A))`
`= ((sin^2 A/cos^2 A))/((1/cos^2 A)) + ((cos^2 A/sin^2 A))/((1/(sin^2 A)))` (using `sin^2 theta + cos^2 theta = 1`)
On further solving by taking the reciprocal of the denominator, we get,
`(sin^2 A/cos^2 A)/(1/cos^2 A) + (cos^2 A/sin^2 A)/(1/sin^2 A) = ((sin^2 A)/(cos^2 A)) (cos^2 A/1) + (cos^2 A/sin^2 A)(sin^2 A/1)`
`= sin^2 A + cos^2 A` (Using `sin^2 theta + cos^2 theta = 1`)
= 1
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 tanθ + sinθ = m and tanθ – sinθ = n, show that `m^2 – n^2 = 4\sqrt{mn}.`
Prove the following trigonometric identities.
`(sec A - tan A)/(sec A + tan A) = (cos^2 A)/(1 + sin A)^2`
Prove the following identities:
`tan A - cot A = (1 - 2cos^2A)/(sin A cos A)`
Prove the following identities:
cosec4 A – cosec2 A = cot4 A + cot2 A
Prove the following identities:
`(1 + cosA)/(1 - cosA) = tan^2A/(secA - 1)^2`
Prove the following identities:
`sinA/(1 + cosA) = cosec A - cot A`
Prove the following identities:
`(sinAtanA)/(1 - cosA) = 1 + secA`
If 4 cos2 A – 3 = 0, show that: cos 3 A = 4 cos3 A – 3 cos A
Write the value of `cosec^2 theta (1+ cos theta ) (1- cos theta).`
Write True' or False' and justify your answer the following :
The value of \[\cos^2 23 - \sin^2 67\] is positive .
Prove the following identity :
`(cotA + tanB)/(cotB + tanA) = cotAtanB`
Prove the following identity :
`(1 + tan^2A) + (1 + 1/tan^2A) = 1/(sin^2A - sin^4A)`
Prove the following identity :
`cosA/(1 - tanA) + sin^2A/(sinA - cosA) = cosA + sinA`
If x = acosθ , y = bcotθ , prove that `a^2/x^2 - b^2/y^2 = 1.`
Prove that `(cot "A" + "cosec A" - 1)/(cot "A" - "cosec A" + 1) = (1 + cos "A")/sin "A"`
Prove the following identities.
cot θ + tan θ = sec θ cosec θ
If sin θ + sin2 θ = 1 show that: cos2 θ + cos4 θ = 1
Choose the correct alternative:
Which is not correct formula?
Prove that `(sintheta + tantheta)/cos theta` = tan θ(1 + sec θ)
