Need help verifying trigonomic identities?

Question

1) square root(1-sinx/1+sinx)=cosx/(1+sinx)

2) sin8x=8sinxcosx(1-2sin^2x)(1-8sin^2xcos^2x)

Answer 1

1)
√((1 - sinx)/(1 + sinx)) =? cosx/(1 + sinx)
Squaring both sides,
(1 - sinx)/(1 + sinx) =? cos^2x/(1 + sinx)^2
(1 - sinx)/(1 + sinx) =? (1 - sin^2x)/(1 + sinx)^2
(1 - sinx)/(1 + sinx) =? (1 + sinx)(1 - sinx)/(1 + sinx)^2
(1 - sinx)/(1 + sinx) = (1 - sinx)/(1 + sinx)

2)
sin8x =? 8sinxcosx(1 - 2sin^2x)(1 - 8sin^2xcos^2x)
sin 2t = (2sin t cos t)
cos 2t = ((cos t)^2 – (sin t)^2)
sin8t = 2sin 4t cos 4t
sin8t = 2( 2sin 2t cos 2t ) ((cos 2t)^2 – (sin 2t)^2)
sin8t = 2( 2(2sin t cos t)((cos t)^2 – (sin t)^2)) ((((cos t)^2 – (sin t)^2))^2 – ((2sin t cos t))^2)
sin8t = (8sin t cos t)((1 – 2(sin t)^2)*(((cos t)^2cos t)^2 - 2(cos t)^2(sin t)^2 – (sin t)^2(sin t)^2) – 4(sin t)^2(cos t)^2)
sin8t = (8sin t cos t)((1 – 2(sin t)^2)*(((cos t)^2 - (sin t)^2(cos t)^2 - 2(cos t)^2(sin t)^2 + (sin t)^2 - (sin t)^2(cos t)^2) – 4(sin t)^2(cos t)^2)
sin8x = (8sin t cos t)((1 – 2(sin t)^2)*(((cos t)^2 + (sin t)^2 – 8(sin t)^2(cos t)^2)
sin8x = 8sinxcosx(1 – 2sin^2x)(1 – 8sin^2xcos^2x)
QED

Answer 2

1) Square the RHS, you get:

RHS²
= cos²x/(1+sinx)²
= (1-sin²x) / (1+sinx)² says Pythagoras
= (1+sinx)(1-sinx) / (1+sinx)² diff of squares
= (1-sinx) / (1+sinx)
= LHS²

2) sin8x = 2sin4xcos4x... keep substituting the equalities for sin4x, cos4x, then sin2x and cos2x, and you'll get there.

Answer 3

a million.) sec^4 - tan^4x = (sec^2 - tan^2x) (sec^2 + tan^2x) = (a million + tan^2x - tan^2x) (sec^2 + tan^2x) = sec^2 + tan^2x 2.) cos ? (sec ? - cos ?) =? sin^2 ? cos ?/cos ? - cos^2 ? =? sin^2 ? a million - cos^2 ? = sin^2 ? sin^2 ? + cos^2 ? - cos^2 ? = sin^2 ? sin^2 ? = sin^2 ? 3.) (sinx + cosx)^2 / sin^2x-cos^2x = sin^2x - cos^2 / (sin x - cos x)^2 (sinx + cosx)^2 / (sinx + cosx)(sinx - cosx) = sin^2x - cos^2 / (sin x - cos x)^2 (sinx + cosx) / (sinx - cosx) = sin^2x - cos^2 / (sin x - cos x)^2 (sinx + cosx)(sinx - cosx) / (sinx - cosx)^2 = sin^2x - cos^2 / (sin x - cos x)^2 (sin^2x - cos^2x) / (sinx - cosx)^2 = sin^2x - cos^2 / (sin x - cos x)^2 4.) (cotx - cscx)^2 = a million-cosx / a million+cosx (cot^2x - 2cotxcscx + csc^2x) = a million-cosx / a million+cosx (a million + 2cos^2x/sin^2x - 2cosx/sin^2x ) = a million-cosx / a million+cosx ((sin^2x + 2cos^2x - 2cosx)/sin^2x ) = a million-cosx / a million+cosx ((a million + cos^2x - 2cosx)/(a million - cos^2x ) = a million-cosx / a million+cosx ((a million - cosx)^2/(a million - cosx)( a million + cosx) = a million-cosx / a million+cosx (a million - cosx)/( a million + cosx) = a million-cosx / a million+cosx 5.) (3cos ? - 4 sin ?)^2 + (4cos ? + 3 sin ?)^2 = 25 9cos^2? - 24sin?cos? + 16sin^2? + 16cos^2? + 24sin?cos? + 9sin^2? = 25 9cos^2? + 9sin^2? + 16sin^2? + 16cos^2? = 25 9(cos^2? + sin^2?) + sixteen(sin^2? + cos^2?) = 25 9 + sixteen = 25