Advertisements
Advertisements
Question
If sin θ + cos θ = p and sec θ + cosec θ = q, then prove that q(p2 – 1) = 2p.
Advertisements
Solution
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
RELATED QUESTIONS
Prove the following identities, where the angles involved are acute angles for which the expressions are defined:
`sqrt((1+sinA)/(1-sinA)) = secA + tanA`
Prove the following trigonometric identities
`((1 + sin theta)^2 + (1 + sin theta)^2)/(2cos^2 theta) = (1 + sin^2 theta)/(1 - sin^2 theta)`
Prove the following trigonometric identities.
`(cot A - cos A)/(cot A + cos A) = (cosec A - 1)/(cosec A + 1)`
Prove the following trigonometric identities.
`tan A/(1 + tan^2 A)^2 + cot A/((1 + cot^2 A)) = sin A cos A`
If x = a sec θ cos ϕ, y = b sec θ sin ϕ and z = c tan θ, show that `x^2/a^2 + y^2/b^2 - x^2/c^2 = 1`
Prove the following identities:
(1 – tan A)2 + (1 + tan A)2 = 2 sec2A
Prove the following identities:
`(1 + cosA)/(1 - cosA) = tan^2A/(secA - 1)^2`
If x = r cos A cos B, y = r cos A sin B and z = r sin A, show that : x2 + y2 + z2 = r2
Show that : `sinA/sin(90^circ - A) + cosA/cos(90^circ - A) = sec A cosec A`
Prove that:
(tan A + cot A) (cosec A – sin A) (sec A – cos A) = 1
`(sec^2 theta-1) cot ^2 theta=1`
`sin theta/((cot theta + cosec theta)) - sin theta /( (cot theta - cosec theta)) =2`
Write the value of `( 1- sin ^2 theta ) sec^2 theta.`
Write the value of `cosec^2 theta (1+ cos theta ) (1- cos theta).`
Write the value of `3 cot^2 theta - 3 cosec^2 theta.`
If a cos θ + b sin θ = 4 and a sin θ − b sin θ = 3, then a2 + b2 =
Prove the following identity :
`cosA/(1 - tanA) + sinA/(1 - cotA) = sinA + cosA`
If x = r sinA cosB , y = r sinA sinB and z = r cosA , prove that `x^2 + y^2 + z^2 = r^2`
Prove that:
`(cot A - 1)/(2 - sec^2 A) = cot A/(1 + tan A)`
Prove that: `(sec θ - tan θ)/(sec θ + tan θ ) = 1 - 2 sec θ.tan θ + 2 tan^2θ`
Prove that sin (90° - θ) cos (90° - θ) = tan θ. cos2θ.
Prove that:
`(sin A + cos A)/(sin A - cos A) + (sin A - cos A)/(sin A + cos A) = 2/(2 sin^2 A - 1)`
Prove that: sin6θ + cos6θ = 1 - 3sin2θ cos2θ.
If x = a tan θ and y = b sec θ then
If a cos θ – b sin θ = c, then prove that (a sin θ + b cos θ) = `± sqrt("a"^2 + "b"^2 -"c"^2)`
Choose the correct alternative:
1 + cot2θ = ?
Prove that `(sin^2theta)/(cos theta) + cos theta` = sec θ
If 1 + sin2θ = 3 sin θ cos θ, then prove that tan θ = 1 or `1/2`.
Simplify (1 + tan2θ)(1 – sinθ)(1 + sinθ)
(sec2 θ – 1) (cosec2 θ – 1) is equal to ______.
