Advertisements
Advertisements
प्रश्न
Prove the following trigonometric identities.
`[tan θ + 1/cos θ]^2 + [tan θ - 1/cos θ]^2 = 2((1 + sin^2 θ)/(1 - sin^2 θ))`
उत्तर १
LHS = (tan θ + sec θ)2 + (tan θ - sec θ)2
`"LHS" = tan^2 θ + sec^2 θ + cancel(2 tan θ. sec θ) + tan^2 θ + sec^2 θ - cancel(2 tan θ. sec θ) ...{(a^2 + b^2 = a^2 + 2ab + b^2),(a^2 - b^2 = a^2 - 2ab + b^2):}`
LHS = `2 tan^2θ + 2 sec^2θ`
LHS = `2[tan^2θ + sec^2θ]`
LHS = `2[sin^2 θ/cos^2 θ + 1/cos^2 θ]`
LHS = `2((sin^2 θ + 1)/cos^2 θ)`
LHS = `2((1 + sin^2 θ)/(1 - sin^2θ)) ...{(sin^2θ + cos^2θ = 1),(∴ cos^2θ = 1 - sin^2θ):}`
RHS = `2((1 + sin^2 θ)/(1 - sin^2θ))`
LHS = RHS
उत्तर २
LHS = `[tan θ + 1/cos θ]^2 + [tan θ - 1/cos θ]^2 ...{(a + b)^2 + (a - b)^2 = 2(a^2 + b^2)}`
LHS = `2[tan^2θ + 1/cos^2θ]`
LHS = `2[sin^2 θ/cos^2 θ + 1/cos^2 θ]`
LHS = `2((sin^2 θ + 1)/cos^2 θ)`
LHS = `2((1 + sin^2 θ)/(1 - sin^2θ)) ...{(sin^2θ + cos^2θ = 1),(∴ cos^2θ = 1 - sin^2θ):}`
RHS = `2((1 + sin^2 θ)/(1 - sin^2θ))`
LHS = RHS
APPEARS IN
संबंधित प्रश्न
Prove the following identities:
`(i) 2 (sin^6 θ + cos^6 θ) –3(sin^4 θ + cos^4 θ) + 1 = 0`
`(ii) (sin^8 θ – cos^8 θ) = (sin^2 θ – cos^2 θ) (1 – 2sin^2 θ cos^2 θ)`
`(1+tan^2A)/(1+cot^2A)` = ______.
Prove that `cosA/(1+sinA) + tan A = secA`
Prove the following trigonometric identities.
`(1 + cos theta - sin^2 theta)/(sin theta (1 + cos theta)) = cot theta`
Prove the following trigonometric identities.
if `T_n = sin^n theta + cos^n theta`, prove that `(T_3 - T_5)/T_1 = (T_5 - T_7)/T_3`
Prove the following trigonometric identities.
`(cos A cosec A - sin A sec A)/(cos A + sin A) = cosec A - sec A`
Prove the following trigonometric identities.
`tan A/(1 + tan^2 A)^2 + cot A/((1 + cot^2 A)) = sin A cos A`
Prove the following identities:
`1 - cos^2A/(1 + sinA) = sinA`
Prove the following identities:
`(sinA - cosA + 1)/(sinA + cosA - 1) = cosA/(1 - sinA)`
`sin^2 theta + cos^4 theta = cos^2 theta + sin^4 theta`
`(sectheta- tan theta)/(sec theta + tan theta) = ( cos ^2 theta)/( (1+ sin theta)^2)`
Write the value of `( 1- sin ^2 theta ) sec^2 theta.`
If 3 `cot theta = 4 , "write the value of" ((2 cos theta - sin theta))/(( 4 cos theta - sin theta))`
If tan A =` 5/12` , find the value of (sin A+ cos A) sec A.
9 sec2 A − 9 tan2 A is equal to
Prove the following identity :
`(1 + cosA)/(1 - cosA) = tan^2A/(secA - 1)^2`
If sec θ = x + `1/(4"x"), x ≠ 0,` find (sec θ + tan θ)
If cosθ + sinθ = `sqrt2` cosθ, show that cosθ - sinθ = `sqrt2` sinθ.
Prove that `sqrt((1 + cos A)/(1 - cos A)) = (tan A + sin A)/(tan A. sin A)`
Prove the following:
(sin α + cos α)(tan α + cot α) = sec α + cosec α
