Advertisements
Advertisements
प्रश्न
If `( cos theta + sin theta) = sqrt(2) sin theta , " prove that " ( sin theta - cos theta ) = sqrt(2) cos theta`
Advertisements
उत्तर
Given : `cos theta + sin theta = sqrt(2) sin theta`
We have `( sin theta + cos theta )^2 + (sin theta - cos theta )^2 =2(sin^2 theta + cos^2 theta )`
`= > ( sqrt(2) sin theta )^2 + ( sin theta - cos theta ) ^2 = 2 `
`= > 2 sin^2 theta + ( sin theta - cos theta ) ^2 = 2`
`= > ( sin theta - cos theta ) ^2 = 2-2 sin^2 theta `
`= > ( sin theta - cos theta ) ^2 =2(1- sin^2 theta)`
`= > ( sin theta - cos theta ) ^2 = 2 cos^2 theta`
`= > ( sin theta - cos theta ) = sqrt(2) cos theta`
Hence proved.
APPEARS IN
संबंधित प्रश्न
If (secA + tanA)(secB + tanB)(secC + tanC) = (secA – tanA)(secB – tanB)(secC – tanC) prove that each of the side is equal to ±1. We have,
As observed from the top of an 80 m tall lighthouse, the angles of depression of two ships on the same side of the lighthouse of the horizontal line with its base are 30° and 40° respectively. Find the distance between the two ships. Give your answer correct to the nearest meter.
Prove the following trigonometric identities.
if x = a cos^3 theta, y = b sin^3 theta` " prove that " `(x/a)^(2/3) + (y/b)^(2/3) = 1`
Prove the following identities:
`cosecA + cotA = 1/(cosecA - cotA)`
Prove that:
`(cosecA - sinA)(secA - cosA) = 1/(tanA + cotA)`
If a cos `theta + b sin theta = m and a sin theta - b cos theta = n , "prove that "( m^2 + n^2 ) = ( a^2 + b^2 )`
Find the value of `(cos 38° cosec 52°)/(tan 18° tan 35° tan 60° tan 72° tan 55°)`
Write the value of \[\cot^2 \theta - \frac{1}{\sin^2 \theta}\]
2 (sin6 θ + cos6 θ) − 3 (sin4 θ + cos4 θ) is equal to
9 sec2 A − 9 tan2 A is equal to
Prove the following identity :
`(sinA + cosA)/(sinA - cosA) + (sinA - cosA)/(sinA + cosA) = 2/(2sin^2A - 1)`
Prove the following identity :
`(tanθ + sinθ)/(tanθ - sinθ) = (secθ + 1)/(secθ - 1)`
Evaluate:
`(tan 65^circ)/(cot 25^circ)`
Prove that: 2(sin6 θ + cos6 θ) – 3 (sin4 θ + cos4 θ) + 1 = 0.
Prove that `sqrt((1 + sin θ)/(1 - sin θ))` = sec θ + tan θ.
Prove that cos θ sin (90° - θ) + sin θ cos (90° - θ) = 1.
If (sin α + cosec α)2 + (cos α + sec α)2 = k + tan2α + cot2α, then the value of k is equal to
If 5x = sec θ and `5/x` = tan θ, then `x^2 - 1/x^2` is equal to
Complete the following activity to prove:
cotθ + tanθ = cosecθ × secθ
Activity: L.H.S. = cotθ + tanθ
= `cosθ/sinθ + square/cosθ`
= `(square + sin^2theta)/(sinθ xx cosθ)`
= `1/(sinθ xx cosθ)` ....... ∵ `square`
= `1/sinθ xx 1/cosθ`
= `square xx secθ`
∴ L.H.S. = R.H.S.
If cot θ = `40/9`, find the values of cosec θ and sinθ,
We have, 1 + cot2θ = cosec2θ
1 + `square` = cosec2θ
1 + `square` = cosec2θ
`(square + square)/square` = cosec2θ
`square/square` = cosec2θ ......[Taking root on the both side]
cosec θ = `41/9`
and sin θ = `1/("cosec" θ)`
sin θ = `1/square`
∴ sin θ = `9/41`
The value is cosec θ = `41/9`, and sin θ = `9/41`
