how do you start this equation i've been tryng it for 20min.
sec^6x(secxtanx)-sec^4x(secxtanx)=sec^5xtan^3x
ec^6x(secxtanx)-sec^4x(secxtanx)=sec^5xtan^3x
Factor out a sec^5 tan and divide thru. Left is
sec^2 x = Tan^2 x
Then this should reduce to
sin^2 x = cos^4 x
take the square root of each side
sin x= cos^2 x
change the cos squared to 1-sin^2
sin^2 x + sin x -1=0
Now you have a quadratic, use the quadratic equation.
To solve the equation sec^6x(secxtanx)-sec^4x(secxtanx)=sec^5xtan^3x, we can follow these steps:
Step 1: Simplify the equation by factoring out sec^5xtanx on the left side:
sec^5xtanx(secxtanx) - sec^4x(secxtanx) = sec^5xtan^3x
Step 2: Divide both sides of the equation by secxtanx:
sec^5xtanx - sec^4x = sec^5xtan^2x
Step 3: Cancel out the sec^5xtanx term on both sides:
- sec^4x = sec^5xtan^2x
Step 4: Rewrite the equation using trigonometric identities:
(sec^2x)(- sec^2x) = (sec^2x)(tan^2x)
Step 5: Simplify the equation further:
-sec^4x = sec^2xtan^2x
Step 6: Convert sec^4x to its equivalent in terms of sin and cos:
- (1/cos^4x) = (1/cos^2x)(sin^2x)
Step 7: Multiply both sides of the equation by cos^4x:
-1 = sin^2xcos^2x
Step 8: Use the Pythagorean Identity sin^2x + cos^2x = 1 to rewrite the equation:
-1 = (1 - cos^2x)cos^2x
Step 9: Distribute and rearrange the equation:
-1 = cos^2x - cos^4x
Step 10: Move all the terms to one side of the equation to form a quadratic equation:
cos^4x - cos^2x -1 = 0
At this point, we have a quadratic equation in terms of cosx. To solve this equation, you can use the quadratic formula or factoring. I will show you how to use the quadratic formula:
Step 11: Apply the quadratic formula:
cosx = [-b ± √(b^2 - 4ac)] / (2a)
For the equation cos^4x - cos^2x - 1 = 0, we have:
a = 1, b = -1, c = -1
cosx = [-(-1) ± √((-1)^2 - 4(1)(-1))] / (2(1))
Simplifying this expression will give you the values of cosx. Remember that cosx can range from -1 to 1, so you should check each solution within that range to find the valid solutions for x.
Note: If you want the exact values of cosx, you may need to further simplify the expression and use trigonometric identities or a calculator to find the values.