Factor and if possible simplify the expression 1 2 sin2x s

Factor and, if possible, simplify the expression. 1 - 2 sin^2x + sin^4x cos^4x (1 + tan^2x) sin^2x (1-sin^2x) ______ Use identities to find the exact value of the trigonometric function. Find sin alpha, given that cos alpha = 2/5 and 0

Solution

given

1 - 2sin^2 ( x) + sin^4 ( x) = 0

1 - 2sin^2 ( x) + sin^4(x) ==== ( a-b)^2 = a^2 -2a b + b^2

( 1 - sin^2(x))^2

(cos^2(x))^2

==> cos^4(x)

------------

( 4)

cos = 2/5 0 < < pi/2

we know

cos =adj side / hyp side ==> 2/5

by pythogorous theorem

hyp^2 = adj^2 + opp^2

5^2 = 2^2 + opp^2

opp^2 ==> 25 - 4

opp= sqrt(21)

sin = opp/ hyp

==> sqrt(21) / 5 in given interval of , sin and cos are positive bcz it is I quadrant