Advertisements
Advertisements
प्रश्न
`(cos^3 θ + sin^3 θ)/(cos θ + sin θ) + (cos ^3 θ - sin^3 θ)/(cos θ - sin θ) = 2`
Prove the following:
`(sin^3 θ + cos^3 θ)/(sin θ +cos θ) + (sin^3 θ - cos ^3 θ)/(sin θ - cos θ) = 2`
Advertisements
उत्तर
LHS= `(cos^3 θ + sin^3 θ)/(cos θ + sin θ) + (cos ^3 θ - sin^3 θ)/(cos θ - sin θ) `
=` ((cos θ + sin θ)(cos^2 θ - cos θ sin θ + sin^2 θ))/((cos θ + sin θ)) + ((cos θ - sin θ)(cos^2 θ + cos θ sin θ + sin^2 θ))/((cos θ - sin θ))`
= (cos2 θ + sin2 θ − cos θ sin θ) + (cos2 θ + sin2 θ + cos θ sin θ)`
= (1 − cos θ sin θ) + (1 + cos θ sin θ)
= 2
= RHS
Hence, LHS = RHS
संबंधित प्रश्न
Prove the following identities:
`(i) (sinθ + cosecθ)^2 + (cosθ + secθ)^2 = 7 + tan^2 θ + cot^2 θ`
`(ii) (sinθ + secθ)^2 + (cosθ + cosecθ)^2 = (1 + secθ cosecθ)^2`
`(iii) sec^4 θ– sec^2 θ = tan^4 θ + tan^2 θ`
If cosθ + sinθ = √2 cosθ, show that cosθ – sinθ = √2 sinθ.
`"If "\frac{\cos \alpha }{\cos \beta }=m\text{ and }\frac{\cos \alpha }{\sin \beta }=n " show that " (m^2 + n^2 ) cos^2 β = n^2`
Prove that ` \frac{\sin \theta -\cos \theta +1}{\sin\theta +\cos \theta -1}=\frac{1}{\sec \theta -\tan \theta }` using the identity sec2 θ = 1 + tan2 θ.
Prove the following trigonometric identities.
tan2θ cos2θ = 1 − cos2θ
Prove the following trigonometric identities.
`tan theta - cot theta = (2 sin^2 theta - 1)/(sin theta cos theta)`
Prove the following identities:
`(1 + cosA)/(1 - cosA) = tan^2A/(secA - 1)^2`
Prove the following identities:
`(sinA + cosA)/(sinA - cosA) + (sinA - cosA)/(sinA + cosA) = 2/(2sin^2A - 1)`
If x = a cos θ and y = b cot θ, show that:
`a^2/x^2 - b^2/y^2 = 1`
If 2 sin A – 1 = 0, show that: sin 3A = 3 sin A – 4 sin3 A
`(1 + cot^2 theta ) sin^2 theta =1`
`(tan theta)/((sec theta -1))+(tan theta)/((sec theta +1)) = 2 sec theta`
Write the value of `( 1- sin ^2 theta ) sec^2 theta.`
cos4 A − sin4 A is equal to ______.
Prove the following identity :
sinθcotθ + sinθcosecθ = 1 + cosθ
Prove the following identity:
tan2A − sin2A = tan2A · sin2A
Prove the following identities:
`(tan"A"+tan"B")/(cot"A"+cot"B")=tan"A"tan"B"`
Prove the following identity :
`1/(sinA + cosA) + 1/(sinA - cosA) = (2sinA)/(1 - 2cos^2A)`
If sinA + cosA = m and secA + cosecA = n , prove that n(m2 - 1) = 2m
If sec θ = `25/7`, then find the value of tan θ.
Prove that:
tan (55° + x) = cot (35° – x)
Find the value of ( sin2 33° + sin2 57°).
Prove that:
`sqrt((sectheta - 1)/(sec theta + 1)) + sqrt((sectheta + 1)/(sectheta - 1)) = 2cosectheta`
Prove that sec2 (90° - θ) + tan2 (90° - θ) = 1 + 2 cot2 θ.
Prove that sin( 90° - θ ) sin θ cot θ = cos2θ.
Prove that `"cosec" θ xx sqrt(1 - cos^2θ) = 1`.
If tan θ × A = sin θ, then A = ?
Prove that `(sec A)/(tan A + cot A) = sin A`.
Prove that cosec θ – cot θ = `(sin θ)/(1 + cos θ)`.
Prove that `(cot A)/(1 - tan A) + (tan A)/(1 - cot A) = 1 + tan A + cot A = sec A . "cosec" A + 1`.
