Advertisements
Advertisements
Question
2 (sin6 θ + cos6 θ) − 3 (sin4 θ + cos4 θ) is equal to
Options
0
1
−1
None of these
Advertisements
Solution
The given expression is `2(sin^6θ+cos^6θ)-3(sin^4θ+cos^4θ)`
Simplifying the given expression, we have
`2(sinθ+cos^6θ)-3(sin^4θ+cos^4θ)`
= `2sin^6θ+2cos^6θ-3sin^4θ-3cos^4θ`
=`(2 sin^6 θ-3sin^4θ)+(2 cos^6-3 cos^4θ)`
=`sin^4θ(2sin^2θ-3)+cos^4θ(2 cos^2θ-3)`
`=sin^4θ{2(1-cos^2)-3}+cos^4θ{2(1-sin^2 θ)-3)`
`= sin^4θ(2-2cos^2θ-3)+cos^4θ(2-2sin^2 θ-3) `
`=sin^4θ(-1-2cos^θ)+cos^4θ(1-2sin^2θ)`
`= -sin^4θ-2 sin^4θ cos^2θ-cos^4θ-2cos^4 θ sin^2θ`
`=sin^4θ-cos^4θ-2 cos^4 θ sin^2θ-2 sin^4 θcos^2θ`
`=-sin^4θ-cos^4θ-2cos^2θ sin^2(cos^2+sin^2θ)`
`=-sin^4θ-cos^4θ-2cos^2θsin^2θ(1)`
`=-sin^4θ-cos^4θ-2cos^2sin^2θ`
`=(sin^4θ+cos^4 θ+2 cos^2 θ sin^2 θ)`
`=-{(sin^2θ)^2+(cos^2θ)^2+2 sin^2 θ cos^2θ}`
` =-(sin^2θ+cos^2θ)^2`
`=-(1)^2`
`=-1`
APPEARS IN
RELATED QUESTIONS
Prove that sin6θ + cos6θ = 1 – 3 sin2θ. cos2θ.
If sinθ + sin2 θ = 1, prove that cos2 θ + cos4 θ = 1
Prove the following trigonometric identities.
`"cosec" theta sqrt(1 - cos^2 theta) = 1`
Prove the following trigonometric identity:
`sqrt((1 + sin A)/(1 - sin A)) = sec A + tan A`
Prove the following trigonometric identities.
`(1 + cot A + tan A)(sin A - cos A) = sec A/(cosec^2 A) - (cosec A)/sec^2 A = sin A tan A - cos A cot A`
Prove the following trigonometric identities
If x = a sec θ + b tan θ and y = a tan θ + b sec θ, prove that x2 − y2 = a2 − b2
Prove the following identities:
`cot^2A/(cosecA + 1)^2 = (1 - sinA)/(1 + sinA)`
If `cosA/cosB = m` and `cosA/sinB = n`, show that : (m2 + n2) cos2 B = n2.
Prove the following identities:
`((cosecA - cotA)^2 + 1)/(secA(cosecA - cotA)) = 2cotA`
If `( cosec theta + cot theta ) =m and ( cosec theta - cot theta ) = n, ` show that mn = 1.
If `cosec theta = 2x and cot theta = 2/x ," find the value of" 2 ( x^2 - 1/ (x^2))`
Write True' or False' and justify your answer the following:
\[ \cos \theta = \frac{a^2 + b^2}{2ab}\]where a and b are two distinct numbers such that ab > 0.
Prove the following identity :
`(1 + sinA)/(1 - sinA) = (cosecA + 1)/(cosecA - 1)`
Prove the following identity :
`sec^4A - sec^2A = sin^2A/cos^4A`
Prove the following identity :
`(cotA + cosecA - 1)/(cotA - cosecA + 1) = (cosA + 1)/sinA`
Prove the following identity :
`(tanθ + sinθ)/(tanθ - sinθ) = (secθ + 1)/(secθ - 1)`
If x = asecθ + btanθ and y = atanθ + bsecθ , prove that `x^2 - y^2 = a^2 - b^2`
Without using trigonometric identity , show that :
`sin(50^circ + θ) - cos(40^circ - θ) = 0`
Prove that `((1 - cos^2 θ)/cos θ)((1 - sin^2θ)/(sin θ)) = 1/(tan θ + cot θ)`
Prove that: `(sin θ - 2sin^3 θ)/(2 cos^3 θ - cos θ) = tan θ`.
