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Hi, I need help with the following questions!

 

1) Given that \(sin(x)+cos(x)=2/5\), what is \(sin^4(x)+cos^4(x) \)?

2) Simplify \(tan(15\pi/2)\)

3) Simplify\(\sqrt{(1-csc^2y)(cos^2y-1)}\).

4) Simplify \(\frac{sin(a)+csc(-a)}{cos(2\pi-a)-sec(a)}-cot(\pi-a)-cot(a)*csc^2(a) \).

5)Simplify\(\frac{sin(-a)*cos(a)+sin^2(-43^{\circ})+cos^2(-403^{\circ})}{cos(720^{\circ}+a)(sec(a)+csc(-a))}*\frac{sin^2(a)-cos^2(a)}{sin^3(a)+cos^3(a)}\).

 

Thanks in advance for the help!!

 Sep 16, 2019
 #1
avatar+104793 
+2

3)   √ [ (1 - csc^2 y) ( cos^2 y - 1) ]

 

√ [ [ (sin^2y - 1) / sin^2 y] [ (-sin^2) ] 

 

 √ [ (sin^2 - 1) *  (-sin^2 y / sin^2y)  ] 

 

√ [ (sin^2 y - 1) (-1)]

 

√ [ 1 - sin^2 y ]  =

 

√ [cos^2 y] =

 

 l cos y  l

 

 

cool cool cool

 Sep 16, 2019
 #2
avatar+104793 
+2

2)  tan (15 pi /2)   =  tan (3pi/2)   = - infinity

 

cool cool cool    

 Sep 16, 2019
 #3
avatar+104793 
+2

1)

sin x + cos x   =  2/5       square both sides

sIn^2 x + 2sin(x)cos(x)  + cos^2x     =  4/25

(sin^2 x + cos^2 x)  + 2sin(x)cos(x)   = 4/25

(1)  + 2sin (x) cos(x)   = 4/25        subtract 1 from  both sides

2sin(x)cos(x)  = 4/25 -1

2sin(x)cos(x)  =  -21/25     divide both sides by 2

sin(x) cos(x)  = -21/50       square both sides

 

sIn^2x cos^2x  =  441/2500

sin^2 x ( 1 - sin^2x)  = 441/2500

sin^2x - sin^4x  = 441/2500

sin^4x  =  sin^2x -441/2500      (1) 

 

sin^2 x cos^2 x  = 441/2500

(1 - cos*2x) (cos^2x)  = 441/250

cos^2x - cos^4x  = 441/2500

cos^4x = cos^2x - 441/2500    (2)

 

Add (1)  and (2)

 

sin^4 x + cos^4x   =   ( sin^2  x   + cos ^2 x )  -  441/2500 - 441/2500

sin^4x + cos ^4 x   =    (1)   -   882/2500

sin^4 x  + cos ^4x     =  [2500 -882 ] / 2500

sin^4 x + cos ^4 x  = 1618 /2500   =  809 / 1250

 

 

cool cool cool

 Sep 16, 2019

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