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 + 2 sin θ cos θ + cos2θ = 3 ...[∵ (a + b)2 = a2 + 2ab + b2]
∴ (sin2θ + cos2θ) + 2 sin θ cos θ = 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 = (sin θ)/(cos θ) + (cos θ)/(sin θ)`
= `(sin^2θ + cos^2θ)/(cos θ sin θ)`
= `1/(sin θ cos θ)` ...[∵ sin2θ + cos2θ = 1]
= `1/1` ...[From (i)]
= 1
APPEARS IN
संबंधित प्रश्न
Prove the following trigonometric identities.
tan2θ cos2θ = 1 − cos2θ
Prove the following trigonometric identities.
`(1 + sin θ)/cos θ+ cos θ/(1 + sin θ) = 2 sec θ`
Prove that:
(sec A − tan A)2 (1 + sin A) = (1 − sin A)
Prove that:
`1/(cosA + sinA - 1) + 1/(cosA + sinA + 1) = cosecA + secA`
If sin A + cos A = m and sec A + cosec A = n, show that : n (m2 – 1) = 2 m
If \[\sin \theta = \frac{4}{5}\] what is the value of cotθ + cosecθ?
If cosec2 θ (1 + cos θ) (1 − cos θ) = λ, then find the value of λ.
Prove that:
(cosec θ - sinθ )(secθ - cosθ ) ( tanθ +cot θ) =1
Prove the following identity :
secA(1 + sinA)(secA - tanA) = 1
Prove the following identity :
`sinA/(1 + cosA) + (1 + cosA)/sinA = 2cosecA`
Prove the following identity :
`(sinA + cosA)/(sinA - cosA) + (sinA - cosA)/(sinA + cosA) = 2/(2sin^2A - 1)`
Prove the following identity :
`(tanθ + 1/cosθ)^2 + (tanθ - 1/cosθ)^2 = 2((1 + sin^2θ)/(1 - sin^2θ))`
Prove the following identity :
`(1 + sinθ)/(cosecθ - cotθ) - (1 - sinθ)/(cosecθ + cotθ) = 2(1 + cotθ)`
If m = a secA + b tanA and n = a tanA + b secA , prove that m2 - n2 = a2 - b2
prove that `1/(1 + cos(90^circ - A)) + 1/(1 - cos(90^circ - A)) = 2cosec^2(90^circ - A)`
If tan θ = 2, where θ is an acute angle, find the value of cos θ.
Prove that `sqrt((1 + sin θ)/(1 - sin θ))` = sec θ + tan θ.
Prove that `(sin θ + tan θ)/(cos θ) = tan θ (1 + sec θ)`.
Let α, β be such that π < α – β < 3π. If sin α + sin β = `-21/65` and cos α + cos β = `-27/65`, then the value of `cos (α - β)/2` is ______.
