Question

sin(2x)-tan(x)=cos(2x)tan(x) Please show steps, thank you!

Answer 1

Answer:

Step-by-step explanation:

We're going to leave the right side alone and work on the left side. In other words we are going to use a series of substitutions for these trig idenitites and get the left side manipulated to look like the right side.

Begin with the fact that sin(2x) = 2sin(x)cos(x) and make that first substitution:

2sin(x)cos(x) - tan(x) = right side

Now use the fact that the tangent is the same as the sin over the cos:

$2sin(x)cos(x)-\frac{sin(x)}{cos(x)}$ = right side

Now find a common denominator of cos(x) by multiplying the 2sin(x)cos(x) by cos(x) and writing the whole mess over that common denominator:

$\frac{2sin(x)cos^2(x)-sin(x)}{cos(x)}$ = right side

Now factor out a sin(x):

$\frac{sin(x)(2cos^2(x)-1)}{cos(x)}$ = right side

If we "split" that up and simplify at the same time, we'll see that sin(x) ovr cos(x) is the same as the tan(x), and that 2cos^2 - 1 is the same as cos(2x):

$\frac{sin(x)}{cos(x)}(2cos^2(x)-1)=tan(x)cos(2x)$ and that the left side now is the same as the right side. You MUST learn to recongize these identities.  I'll attach a copy that I made and give to my pre-calculus and calculus classes every year, if I am able to.