Advertisements
Advertisements
प्रश्न
Prove the following trigonometric identities.
`tan θ/(1 - cot θ) + cot θ/(1 - tan θ) = 1 + tan θ + cot θ`
Prove the following:
`tan θ/(1 - cot θ) + cot θ/(1 - tan θ) = 1 + tan θ + cot θ`
Advertisements
उत्तर
We need to prove `tan θ/(1 - cot θ) + cot θ/(1 - tan θ) = 1 + tan θ + cot θ`
Now using cot θ = `1/tan θ` in the LHS, we get
`tan θ/(1 - cot θ) + cot θ/(1 - tan θ) = tan θ/(1 - 1/tan θ) + (1/tan θ)/(1 - tan θ)`
`= tan θ/(((tan θ - 1)/tan θ)) + 1/(tan θ(1 - tan θ))`
`= (tan θ)/(tan θ - 1)(tan θ) + 1/(tan θ(1 - tan θ)`
`= tan^2 θ/(tan θ - 1) - 1/(tan θ(tan θ - 1))`
`= (tan^3 θ - 1)/(tan θ(tan θ - 1))`
Further using the identity `a^3 - b^3 = (a - b)(a^2 + ab + b^2)`, we get
`(tan^3 θ - 1)/(tan(tan θ - 1)) = ((tan θ - 1)(tan^2 θ + tan θ + 1))/(tan θ (tan θ - 1))`
`= (tan^2 θ + tan θ + 1)/(tan θ)`
`= tan^2 θ/tan θ+ tan θ/tan θ + 1/tan θ`
= tan θ + 1 + cot θ
Hence `tan θ/(1 - cot θ) + cot θ/(1 - tan θ) = 1 + tan θ + cot θ`
APPEARS IN
संबंधित प्रश्न
Prove that:
`sqrt(sec^2A + cosec^2A) = tanA + cotA`
`1+((tan^2 theta) cot theta)/(cosec^2 theta) = tan theta`
`((sin A- sin B ))/(( cos A + cos B ))+ (( cos A - cos B ))/(( sinA + sin B ))=0`
Prove that `( sintheta - 2 sin ^3 theta ) = ( 2 cos ^3 theta - cos theta) tan theta`
Prove the following identities:
`(sec"A"-1)/(sec"A"+1)=(sin"A"/(1+cos"A"))^2`
Prove that:
`(cot A - 1)/(2 - sec^2 A) = cot A/(1 + tan A)`
Prove that `(sec θ - 1)/(sec θ + 1) = ((sin θ)/(1 + cos θ ))^2`
Prove that sec2 (90° - θ) + tan2 (90° - θ) = 1 + 2 cot2 θ.
Prove the following identities.
`(1 - tan^2theta)/(cot^2 theta - 1)` = tan2 θ
If cosec θ + cot θ = p, then prove that cos θ = `(p^2 - 1)/(p^2 + 1)`
