Advertisements
Advertisements
प्रश्न
Prove the following identities.
(sin θ + sec θ)2 + (cos θ + cosec θ)2 = 1 + (sec θ + cosec θ)2
Advertisements
उत्तर
(sin θ + sec θ)2 + (cos θ + cosec θ)2 = 1 + (sec θ + cosec θ)2
L.H.S = [(sin θ + sec θ)2 + (cos θ + cosec θ)2]
= [sin2 θ + sec2 θ + 2 sin θ sec θ + cos2 θ + cosec2 θ + 2 cos θ cosec θ]
= (sin2θ + cos2θ) + (sec2θ + cosec2θ) + 2 (sinθ secθ + cos θ cosec θ)
= `1 + sec^2 theta + "cosec"^2 theta + 2[sin theta xx 1/cos theta + cos theta xx 1/sin theta]`
= `1 + sec^2 theta + "cosec"^2 theta + 2 [(sin^2 theta + cos^2 theta)/(sintheta cos theta)]`
= `1 + sec^2 theta + "cosec"^2 theta + 2 xx 1/(sintheta costheta)`
= 1 + sec2θ + cosec2θ + 2 secθ cosecθ
= 1 + (secθ + cosecθ)2
L.H.S = R.H.S
∴ (sin θ + sec θ)2 + (cos θ + cosec θ)2 = 1 + (sec θ + cosec θ)2
APPEARS IN
संबंधित प्रश्न
Prove the following trigonometric identities.
`1/(sec A - 1) + 1/(sec A + 1) = 2 cosec A cot A`
Prove the following identities:
`sqrt((1 + sinA)/(1 - sinA)) = sec A + tan A`
Prove the following identities:
`(sinA - cosA + 1)/(sinA + cosA - 1) = cosA/(1 - sinA)`
If tanθ `= 3/4` then find the value of secθ.
If a cos θ + b sin θ = m and a sin θ − b cos θ = n, then a2 + b2 =
Prove that `sqrt(2 + tan^2 θ + cot^2 θ) = tan θ + cot θ`.
Without using trigonometric table, prove that
`cos^2 26° + cos 64° sin 26° + (tan 36°)/(cot 54°) = 2`
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 `sec θ + tan θ = sqrt(3)`, complete the activity to find the value of sec θ – tan θ.
Activity:
`square = 1 + tan^2θ` ...[Fundamental trigonometric identity]
`square - tan^2θ = 1`
`(sec θ + tan θ) . (sec θ - tan θ) = square`
`sqrt(3) . (sec θ - tan θ) = 1`
`(sec θ - tan θ) = square`
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 ______.
