Advertisements
Advertisements
Question
Prove that:
Sin4θ - cos4θ = 1 - 2cos2θ
Advertisements
Solution
Sin4θ – cos4θ = 1 – 2cos2θ
LHS = Sin4θ – cos4θ
LHS = (Sin2θ)2 – (cos2θ)2
LHS = (Sin2θ + cos2θ)(Sin2θ - cos2θ) ...[a2 – b2 = (a + b)(a – b)]
LHS = (Sin2θ – cos2θ).(1) ...(Sin2θ + cos2θ = 1)
LHS = 1 – cos2θ – cos2θ ...(1 – Sin2θ = cos2θ)
LHS = 1 – 2cos2θ
RHS = 1 – 2cos2θ
LHS = RHS
Hence proved.
RELATED QUESTIONS
Prove the following trigonometric identities:
`(\text{i})\text{ }\frac{\sin \theta }{1-\cos \theta }=\text{cosec}\theta+\cot \theta `
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.
(sec2 θ − 1) (cosec2 θ − 1) = 1
Prove the following trigonometric identities
sec4 A(1 − sin4 A) − 2 tan2 A = 1
Prove the following identities:
`1/(secA + tanA) = secA - tanA`
If x = r cos A cos B, y = r cos A sin B and z = r sin A, show that : x2 + y2 + z2 = r2
Prove the following identities:
`(1 - cosA)/sinA + sinA/(1 - cosA)= 2cosecA`
(i)` (1-cos^2 theta )cosec^2theta = 1`
`(cot ^theta)/((cosec theta+1)) + ((cosec theta + 1))/cot theta = 2 sec theta`
`(cos^3 θ + sin^3 θ)/(cos θ + sin θ) + (cos ^3 θ - sin^3 θ)/(cos θ - sin θ) = 2`
Write the value of `( 1- sin ^2 theta ) sec^2 theta.`
If `sin theta = x , " write the value of cot "theta .`
2 (sin6 θ + cos6 θ) − 3 (sin4 θ + cos4 θ) is equal to
If x = r sin θ cos ϕ, y = r sin θ sin ϕ and z = r cos θ, then
Prove the following identity :
`sin^2Acos^2B - cos^2Asin^2B = sin^2A - sin^2B`
Prove the following identity :
`sec^2A + cosec^2A = sec^2Acosec^2A`
Prove that `((1 - cos^2 θ)/cos θ)((1 - sin^2θ)/(sin θ)) = 1/(tan θ + cot θ)`
Given that sinθ + 2cosθ = 1, then prove that 2sinθ – cosθ = 2.
Simplify (1 + tan2θ)(1 – sinθ)(1 + sinθ)
Show that `(cos^2(45^circ + θ) + cos^2(45^circ - θ))/(tan(60^circ + θ) tan(30^circ - θ)) = 1`.
