Advertisements
Advertisements
प्रश्न
If sin θ + cos θ = p and sec θ + cosec θ = q, then prove that q(p2 – 1) = 2p.
उत्तर
Given that,
sin θ + cos θ = p ...(i)
and sec θ + cosec θ = q
`\implies 1/cos θ + 1/sin θ` = q ...`[∵ sec θ = 1/cos θ and "cosec" θ = 1/sinθ]`
`\implies (sin θ + cos θ)/(sin θ . cos θ)` = q
`\implies "p"/(sin θ . cos θ)` = q ...[From equation (i)]
`\implies` sin θ. cos θ = `"p"/"q"` ...(ii)
sin θ + cos θ = p
On squaring both sides, we get
(sin θ + cos θ)2 = p2
`\implies` (sin2 θ + cos2 θ) + 2 sin θ . cos θ = p2 ...[∵ (a + b)2 = a2 + 2ab + b2]
`\implies` 1 + 2sin θ . cos θ = p2 ...[∵ sin2 θ + cos2 θ = 1]
`\implies` `1 + 2 . "p"/"q"` = p2 ...[From equation (iii)]
`\implies` q + 2p = p2q
`\implies` 2p = p2q – q
`\implies` q(p2 – 1) = 2p
Hence proved.
APPEARS IN
संबंधित प्रश्न
Prove that sin6θ + cos6θ = 1 – 3 sin2θ. cos2θ.
Show that `sqrt((1+cosA)/(1-cosA)) = cosec A + cot A`
Show that `sqrt((1-cos A)/(1 + cos A)) = sinA/(1 + cosA)`
Prove the following trigonometric identities.
`(1 + cos theta - sin^2 theta)/(sin theta (1 + cos theta)) = cot theta`
If cos θ + cos2 θ = 1, prove that sin12 θ + 3 sin10 θ + 3 sin8 θ + sin6 θ + 2 sin4 θ + 2 sin2 θ − 2 = 1
Prove the following identities:
`1/(tan A + cot A) = cos A sin A`
Prove the following identities:
`(cosecA - 1)/(cosecA + 1) = (cosA/(1 + sinA))^2`
Show that : tan 10° tan 15° tan 75° tan 80° = 1
Prove the following identities:
`cotA/(1 - tanA) + tanA/(1 - cotA) = 1 + tanA + cotA`
Prove that:
`(sinA - cosA)(1 + tanA + cotA) = secA/(cosec^2A) - (cosecA)/(sec^2A)`
`(1 + cot^2 theta ) sin^2 theta =1`
`1+((tan^2 theta) cot theta)/(cosec^2 theta) = tan theta`
cosec4θ − cosec2θ = cot4θ + cot2θ
` (sin theta - cos theta) / ( sin theta + cos theta ) + ( sin theta + cos theta ) / ( sin theta - cos theta ) = 2/ ((2 sin^2 theta -1))`
If `( sin theta + cos theta ) = sqrt(2) , " prove that " cot theta = ( sqrt(2)+1)`.
If tan A = n tan B and sin A = m sin B , prove that `cos^2 A = ((m^2-1))/((n^2 - 1))`
If `cos theta = 7/25 , "write the value of" ( tan theta + cot theta).`
Write True' or False' and justify your answer the following :
The value of the expression \[\sin {80}^° - \cos {80}^°\]
Prove the following identity :
`(cosecA - sinA)(secA - cosA)(tanA + cotA) = 1`
Without using trigonometric table , evaluate :
`(sin47^circ/cos43^circ)^2 - 4cos^2 45^circ + (cos43^circ/sin47^circ)^2`
Find the value of `θ(0^circ < θ < 90^circ)` if :
`cos 63^circ sec(90^circ - θ) = 1`
Prove that sin( 90° - θ ) sin θ cot θ = cos2θ.
If cot θ + tan θ = x and sec θ – cos θ = y, then prove that `(x^2y)^(2/3) – (xy^2)^(2/3)` = 1
If a cos θ – b sin θ = c, then prove that (a sin θ + b cos θ) = `± sqrt("a"^2 + "b"^2 -"c"^2)`
Show that tan 7° × tan 23° × tan 60° × tan 67° × tan 83° = `sqrt(3)`
If 4 tanβ = 3, then `(4sinbeta-3cosbeta)/(4sinbeta+3cosbeta)=` ______.
Given that sin θ = `a/b`, then cos θ is equal to ______.
Show that `(cos^2(45^circ + theta) + cos^2(45^circ - theta))/(tan(60^circ + theta) tan(30^circ - theta))` = 1
(1 – cos2 A) is equal to ______.
