Advertisements
Advertisements
Question
If x sin3 θ + y cos3 θ = sin θ cos θ and x sin θ = y cos θ, then prove that x2 + y2 = 1
Advertisements
Solution
Given x sin2 θ + y cos2 θ = sin θ cos θ
x sin θ = y cos θ ...(1)
x sin3 θ + y cos3 θ = sin θ cos θ
x sin θ (sin2 θ) + y cos θ (cos2 θ) = sin θ cos θ
x sin θ (sin2 θ) + x sin θ (cos2 θ) = sin θ cos θ
x sin θ (sin2 θ + cos2 θ) = sin θ cos θ
x sin θ = sin θ cos θ
x = cos θ
substitute x = cos θ in (1)
cos θ sin θ = y cos θ y = sin θ
L.H.S = x2 + y2 = cos2 θ + sin2 θ = 1
L.H.S = R.H.S
Hence it is proved.
APPEARS IN
RELATED QUESTIONS
If `sec alpha=2/sqrt3` , then find the value of `(1-cosecalpha)/(1+cosecalpha)` where α is in IV quadrant.
Prove that `\frac{\sin \theta -\cos \theta }{\sin \theta +\cos \theta }+\frac{\sin\theta +\cos \theta }{\sin \theta -\cos \theta }=\frac{2}{2\sin^{2}\theta -1}`
Prove the following trigonometric identities.
(cosec θ − sec θ) (cot θ − tan θ) = (cosec θ + sec θ) ( sec θ cosec θ − 2)
If m = a sec A + b tan A and n = a tan A + b sec A, then prove that : m2 – n2 = a2 – b2
Prove the following identities:
`cot^2A((secA - 1)/(1 + sinA)) + sec^2A((sinA - 1)/(1 + secA)) = 0`
Prove that:
cos A (1 + cot A) + sin A (1 + tan A) = sec A + cosec A
If `cos theta = 2/3 , "write the value of" ((sec theta -1))/((sec theta +1))`
Prove the following identity :
`cosA/(1 - tanA) + sinA/(1 - cotA) = sinA + cosA`
Prove that (sin θ + cosec θ)2 + (cos θ + sec θ)2 = 7 + tan2 θ + cot2 θ.
tan2θ – sin2θ = tan2θ × sin2θ. For proof of this complete the activity given below.
Activity:
L.H.S. = `square`
= `square (1 - (sin^2θ)/(tan^2θ))`
= `tan^2θ (1 - square/((sin^2θ)/(cos^2θ)))`
= `tan^2θ (1 - (sin^2θ)/1 xx (cos^2θ)/square)`
= `tan^2θ (1 - square)`
= `tan^2θ xx square` ...[1 – cos2θ = sin2θ]
= R.H.S.
