Show that : `sin^4alpha+cos^4alphage 1/2`

Show that : sin^6 alpha+ cos^6 alpha ge 1/4

show that : sin^2 alpha + sin^2(alpha +( 2pi)/n) +sin^2(alpha +( 4pi)/n) + ..... n terms = n/2

Show that cos ^2 alpha + cos^2 (alpha +Beta) - 2 cos alpha cos betacos (alpha+ beta) =sin^2 beta

Show that 4 sin alpha.sin (alpha + pi/3) sin (alpha + 2pi/3) = sin 3alpha

Q. Prove that from the equality sin^4 alpha/a+cos^4 alpha/b=1/(a+b) follows the relation; sin^8 alpha/a^3+cos^8 alpha/b^3=1/(a+b)^3.

Show that: sin^2 alpha + sin^2 beta + 2sinalpha sinbeta cos(alpha+beta)=sin^2 (alpha+beta)

If pi lt alpha lt (3pi)/2 then find the value of expression sqrt(4 sin^4 alpha+ sin^2 2alpha)+ 4 cos^2(pi/4- alpha/2)

If cos^4 alpha +k and sin^4 alpha+k are the roots of x^2+lambda(2 x+1)=0 and sin^2 alpha+1 and cos^2 alpha +1 are the roots of x^2 +8 x+4=0, then the sum of the possible values of lambda is

Prove that from the equality (sin^(4)alpha)/a+(cos^(4)alpha)/b=1/(a+b) follows the relations, (sin^(8)alpha)/a^(3) +(cos^(8)alpha)/b^(3)=1/(a+b)^(3)

Prove that (sin alpha cos beta + cos alpha sin beta) ^(2) + (cos alpha coa beta - sin alpha sin beta) ^(2) =1.