Advertisements
Advertisements
प्रश्न
`1/sin^2θ - 1/cos^2θ - 1/tan^2θ - 1/cot^2θ - 1/sec^2θ - 1/("cosec"^2θ) = -3`, then find the value of θ.
Advertisements
उत्तर
`1/sin^2θ - 1/cos^2θ - 1/tan^2θ - 1/cot^2θ - 1/sec^2θ - 1/("cosec"^2θ) = -3`
cosec2θ − sec2θ − cot2θ − tan2θ − cos2θ − sin2θ = −3 ...`[because sintheta = 1/(cosectheta), costheta = 1/(sectheta), tantheta = 1/(cottheta)]`
⇒ 1 + cot2θ − 1 − tan2θ − cot2θ − tan2θ − 1 = −3
⇒ − 2 tan2θ − 1 = − 3 ...`[(because 1 + cot^2theta = cosec^2theta), (1 + sec^2theta = tan^2theta), (sin^2theta + cos^2theta = 1)]`
⇒ −2 tan2θ = − 3 + 1
⇒ −2 tan2θ = −2
⇒ tan2θ = 1
⇒ tan θ = 1 ...(Taking square root on both sides)
⇒ tan θ = tan 45°
∴ θ = 45°
APPEARS IN
संबंधित प्रश्न
If m=(acosθ + bsinθ) and n=(asinθ – bcosθ) prove that m2+n2=a2+b2
Prove the following trigonometric identities.
`(1 - tan^2 A)/(cot^2 A -1) = tan^2 A`
Prove the following identities:
(cosec A – sin A) (sec A – cos A) (tan A + cot A) = 1
Prove the following identities:
`(cotA - cosecA)^2 = (1 - cosA)/(1 + cosA)`
Prove the following identities:
`sqrt((1 - cosA)/(1 + cosA)) = cosec A - cot A`
Prove the following identities:
`((cosecA - cotA)^2 + 1)/(secA(cosecA - cotA)) = 2cotA`
Prove the following identities:
sec4 A (1 – sin4 A) – 2 tan2 A = 1
cos4 A − sin4 A is equal to ______.
Prove the following identities:
`(tan"A"+tan"B")/(cot"A"+cot"B")=tan"A"tan"B"`
Prove the following identity :
`(cotA + cosecA - 1)/(cotA - cosecA + 1) = (cosA + 1)/sinA`
Find the value of sin 30° + cos 60°.
Find the value of ( sin2 33° + sin2 57°).
Prove that: `sqrt((1 - cos θ)/(1 + cos θ)) = "cosec" θ - cot θ`.
If tan α = n tan β, sin α = m sin β, prove that cos2 α = `(m^2 - 1)/(n^2 - 1)`.
If cosθ + sinθ = `sqrt2` cosθ, show that cosθ - sinθ = `sqrt2` sinθ.
Prove that cot2θ × sec2θ = cot2θ + 1.
To prove cot θ + tan θ = cosec θ × sec θ, complete the activity given below.
Activity:
L.H.S. = `square`
= `square/(sinθ) + (sinθ)/(cosθ)`
= `(cos^2θ + sin^2θ)/square`
= `1/(sinθ.cosθ)` ...`[cos^2θ + sin^2θ = square]`
= `1/(sinθ) xx 1/square`
= `square`
= R.H.S.
Let x1, x2, x3 be the solutions of `tan^-1((2x + 1)/(x + 1)) + tan^-1((2x - 1)/(x - 1))` = 2tan–1(x + 1) where x1 < x2 < x3 then 2x1 + x2 + x32 is equal to ______.
Eliminate θ if x = r cosθ and y = r sinθ.
(sec2 θ – 1) (cosec2 θ – 1) is equal to ______.
