Advertisements
Advertisements
प्रश्न
If sin θ + cos θ = `sqrt(3)`, then show that tan θ + cot θ = 1
Advertisements
उत्तर
sin θ + cos θ = `sqrt(3)` ......[Given]
∴ (sin θ + cos θ)2 = 3 ......[Squaring on both sides]
∴ sin2θ + 2sinθ cosθ + cos2θ = 3 ......[∵ (a + b)2 = a2 + 2ab + b2]
∴ (sin2θ + cos2θ) + 2sinθ cosθ = 3
∴ 1 + 2 sin θ cos θ = 3 ......[∵ sin2θ + cos2θ = 1]
∴ 2 sin θ cos θ = 2
∴ sin θ cos θ = 1 ......(i)
tan θ + cot θ = `sintheta/costheta + costheta/sintheta`
= `(sin^2theta + cos^2theta)/(costhetasintheta)`
= `1/(sintheta costheta)` ......[∵ sin2θ + cos2θ = 1]
= `1/1` ......[From (i)]
= 1
APPEARS IN
संबंधित प्रश्न
Evaluate without using trigonometric tables:
`cos^2 26^@ + cos 64^@ sin 26^@ + (tan 36^@)/(cot 54^@)`
Prove the following trigonometric identities.
`"cosec" theta sqrt(1 - cos^2 theta) = 1`
Prove the following trigonometric identities.
`(1 - sin θ)/(1 + sin θ) = (sec θ - tan θ)^2`
Prove the following trigonometric identities.
if x = a cos^3 theta, y = b sin^3 theta` " prove that " `(x/a)^(2/3) + (y/b)^(2/3) = 1`
Prove the following identities:
cot2 A – cos2 A = cos2 A . cot2 A
Prove the following identities:
(sec A – cos A) (sec A + cos A) = sin2 A + tan2 A
Prove the following identities:
`cosecA + cotA = 1/(cosecA - cotA)`
If x = a cos θ and y = b cot θ, show that:
`a^2/x^2 - b^2/y^2 = 1`
If x=a `cos^3 theta and y = b sin ^3 theta ," prove that " (x/a)^(2/3) + ( y/b)^(2/3) = 1.`
If `cos theta = 2/3 , " write the value of" (4+4 tan^2 theta).`
Prove the following identity :
`sec^2A + cosec^2A = sec^2Acosec^2A`
Without using trigonometric identity , show that :
`cos^2 25^circ + cos^2 65^circ = 1`
Without using trigonometric identity , show that :
`sec70^circ sin20^circ - cos20^circ cosec70^circ = 0`
Prove that `sqrt((1 + sin θ)/(1 - sin θ))` = sec θ + tan θ.
If tan α = n tan β, sin α = m sin β, prove that cos2 α = `(m^2 - 1)/(n^2 - 1)`.
Prove that : `tan"A"/(1 - cot"A") + cot"A"/(1 - tan"A") = sec"A".cosec"A" + 1`.
Prove that `(tan(90 - theta) + cot(90 - theta))/("cosec" theta)` = sec θ
Prove that sin4A – cos4A = 1 – 2cos2A
Prove that (sec θ + tan θ) (1 – sin θ) = cos θ
