Advertisements
Advertisements
प्रश्न
Prove the following trigonometric identities.
`tan theta/(1 - cot theta) + cot theta/(1 - tan theta) = 1 + tan theta + cot theta`
Advertisements
उत्तर
We need to prove `tan theta/(1 - cot theta) + cot theta/(1 - tan theta) = 1 + tan theta + cot theta`
Now using cot theta = `1/tan theta` in the LHS we get
`tan theta/(1 - cot theta) + cot theta/(1 - tan theta) = tan theta/(1 - 1/tan theta) + (1/tan theta)/(1 - tan theta)`
`= tan theta/(((tan theta - 1)/tan theta)) + 1/(tan theta(1 - tan theta))`
`= (tan theta)/(tan theta - 1)(tan theta) + 1/(tan theta(1 - tan theta)`
`= tan^2 theta/(tan theta - 1) - 1/(tan theta(tan theta - 1))`
`= (tan^3 theta - 1)/(tan theta(tan theta - 1))`
Further using the identity `a^3 - b^3 = (a - b)(a^2 + ab + b^2)`, we get
`(tan^3 theta - 1)/(tan(tan theta - 1)) = ((tan theta - 1)(tan^2 theta + tan theta + 1))/(tan theta (tan theta - 1))`
`= (tan^2 theta + tan theta + 1)/(tan theta)`
`= tan^2 theta/tan theta + tan theta/tan theta + 1/tan theta`
`= tan theta + 1 + cot theta`
Hence `tan theta/(1 - cot theta) + cot theta/(1 - tan theta) = 1 + tan theta + cot theta`
APPEARS IN
संबंधित प्रश्न
(1 + tan θ + sec θ) (1 + cot θ − cosec θ) = ______.
Prove the following trigonometric identities.
`(cos A cosec A - sin A sec A)/(cos A + sin A) = cosec A - sec A`
If 3 sin θ + 5 cos θ = 5, prove that 5 sin θ – 3 cos θ = ± 3.
Prove the following identities:
`tan A - cot A = (1 - 2cos^2A)/(sin A cos A)`
Prove the following identities:
`(secA - tanA)/(secA + tanA) = 1 - 2secAtanA + 2tan^2A`
Prove the following identities:
`cosA/(1 + sinA) + tanA = secA`
`(1 + cot^2 theta ) sin^2 theta =1`
Write the value of cos1° cos 2°........cos180° .
Define an identity.
If sec θ + tan θ = x, write the value of sec θ − tan θ in terms of x.
9 sec2 A − 9 tan2 A is equal to
Prove the following identity :
`sqrt((secq - 1)/(secq + 1)) + sqrt((secq + 1)/(secq - 1))` = 2 cosesq
Without using trigonometric identity , show that :
`cos^2 25^circ + cos^2 65^circ = 1`
Prove that: `(sec θ - tan θ)/(sec θ + tan θ ) = 1 - 2 sec θ.tan θ + 2 tan^2θ`
Prove that sin θ sin( 90° - θ) - cos θ cos( 90° - θ) = 0
Prove that `((1 - cos^2 θ)/cos θ)((1 - sin^2θ)/(sin θ)) = 1/(tan θ + cot θ)`
Prove that `(1 + sintheta)/(1 - sin theta)` = (sec θ + tan θ)2
If cos A = `(2sqrt("m"))/("m" + 1)`, then prove that cosec A = `("m" + 1)/("m" - 1)`
If cosec A – sin A = p and sec A – cos A = q, then prove that `("p"^2"q")^(2/3) + ("pq"^2)^(2/3)` = 1
(1 – cos2 A) is equal to ______.
