Advertisements
Advertisements
प्रश्न
Prove the following trigonometric identities.
`(tan^3 theta)/(1 + tan^2 theta) + (cot^3 theta)/(1 + cot^2 theta) = sec theta cosec theta - 2 sin theta cos theta`
Advertisements
उत्तर
`(tan^3 theta)/(1 + tan^2 theta) + (cot^3 theta)/(1 + cot^2 theta) ` [`∵ sec^2 theta - tan^2 theta = 1 - cosec^2 theta - cot^2 theta = 1`]
`= tan theta + cos^2 theta = cot^3 theta xx sin^3 theta`
`[∵ 1/sec^2 theta = cos^2 theta, 1/cosec^2 theta = 1 + cot^2 theta]`
`sin^3 theta/cos^3 theta xx cos^2 theta + cos^3 theta/sin^3 theta xx sin^2 theta`
`sin^3 theta/cos theta + cos^3 theta/sin theta`
`= (sin^4 theta + cos^4 theta)/(sin theta cos theta)`
` (1 - 2sin^2 theta cos^2 theta)/(sin theta cos theta)`
`1/(sin theta cos theta) - (2 sin^2 theta cos^2 theta)/(sin theta cos theta)`
`sec theta cosec theta - 2sin theta cos theta`.
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.`
Prove the following trigonometric identities.
(cosec θ − sec θ) (cot θ − tan θ) = (cosec θ + sec θ) ( sec θ cosec θ − 2)
Prove the following trigonometric identities.
`(cot A + tan B)/(cot B + tan A) = cot A tan B`
Prove the following identities:
`1/(tan A + cot A) = cos A sin A`
Prove the following identities:
`cotA/(1 - tanA) + tanA/(1 - cotA) = 1 + tanA + cotA`
Prove the following identities:
(1 + tan A + sec A) (1 + cot A – cosec A) = 2
If a cos `theta + b sin theta = m and a sin theta - b cos theta = n , "prove that "( m^2 + n^2 ) = ( a^2 + b^2 )`
If `cos theta = 2/3 , " write the value of" (4+4 tan^2 theta).`
If `cot theta = 1/ sqrt(3) , "write the value of" ((1- cos^2 theta))/((2 -sin^2 theta))`
(cosec θ − sin θ) (sec θ − cos θ) (tan θ + cot θ) is equal to
The value of sin2 29° + sin2 61° is
Prove the following identity :
`(cosA + sinA)^2 + (cosA - sinA)^2 = 2`
Prove the following identity :
`(cotA + cosecA - 1)/(cotA - cosecA + 1) = (cosA + 1)/sinA`
`(sin A)/(1 + cos A) + (1 + cos A)/(sin A)` = 2 cosec A
Prove that : `(sin(90° - θ) tan(90° - θ) sec (90° - θ))/(cosec θ. cos θ. cot θ) = 1`
Prove that `(sin (90° - θ))/cos θ + (tan (90° - θ))/cot θ + (cosec (90° - θ))/sec θ = 3`.
Choose the correct alternative:
sin θ = `1/2`, then θ = ?
If tan θ = `7/24`, then to find value of cos θ complete the activity given below.
Activity:
sec2θ = 1 + `square` ......[Fundamental tri. identity]
sec2θ = 1 + `square^2`
sec2θ = 1 + `square/576`
sec2θ = `square/576`
sec θ = `square`
cos θ = `square` .......`[cos theta = 1/sectheta]`
`(cos^2 θ)/(sin^2 θ) - 1/(sin^2 θ)`, in simplified form, is ______.
Find the value of sin2θ + cos2θ
Solution:
In Δ ABC, ∠ABC = 90°, ∠C = θ°
AB2 + BC2 = `square` .....(Pythagoras theorem)
Divide both sides by AC2
`"AB"^2/"AC"^2 + "BC"^2/"AC"^2 = "AC"^2/"AC"^2`
∴ `("AB"^2/"AC"^2) + ("BC"^2/"AC"^2) = 1`
But `"AB"/"AC" = square and "BC"/"AC" = square`
∴ `sin^2 theta + cos^2 theta = square`
