prove the identity:
(cosx)(tanx + sinx cotx)=sinx+cos(squared)x
i need steps to show how i got the answer
generally, it is a good idea to change all trig ratios to sines and cosines, and start with the more complicated-looking side.
so....
LS =
cosx(sinx/cosx + sinx(cosx/sinx)
=
=
just expand and simplify and the RS pops out
To prove the given trigonometric identity, we start by simplifying both sides.
Left-hand side (LHS):
LHS = (cosx)(tanx + sinx cotx)
Now, let's rewrite tanx and cotx in terms of sinx and cosx:
tanx = sinx/cosx
cotx = cosx/sinx
Substituting these values into the LHS:
LHS = (cosx)(sinx/cosx + sinx(cosx/sinx))
Next, simplify by adding the fractions:
LHS = (cosx)(sinx/cosx + sinx(cosx/sinx))
= (cosx)(sinx/cosx + sinx^2/cosx)
Now, find a common denominator for the two fractions:
LHS = (cosx)((sinx + sinx^2)/cosx)
Simplify by canceling the common term "cosx":
LHS = sinx + sinx^2
Right-hand side (RHS):
RHS = sinx + cos^2x
Now let's compare the LHS and the RHS:
LHS = sinx + sinx^2
RHS = sinx + cos^2x
We can see that the LHS and the RHS are the same expression. Therefore, we have proven the identity.
To summarize the steps:
1. Rewrite tanx and cotx in terms of sinx and cosx.
2. Substitute these values into the LHS of the identity.
3. Simplify by adding the fractions.
4. Find a common denominator for the fractions.
5. Cancel out any common factors.
6. Compare the LHS and the RHS to see if they are the same expression.