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Simplifying Trig functions...urgent help needed
- Thread starter Samara
- Start date
For the last one, are you sure that's the correct equation?.
This this an identity to prove or an equation to solve?. If it's an identity to prove, in general \(\displaystyle csc(x)-cos(x)\neq{sin(x)tan(x)}\)
If it's an equation:
\(\displaystyle csc(x)-cos(x)=sin(x)tan(x)\)
\(\displaystyle \frac{1}{sin(x)}-cos(x)=\frac{sin^{2}(x)}{cos(x)}\)
Multiply through by cos(x):
\(\displaystyle sin^{2}(x)=cot(x)-cos^{2}(x)\)
\(\displaystyle sin^{2}(x)+cos^{2}(x)=cot(x)\)
Now, can you finish?. Notice the left side, what that is?.
This this an identity to prove or an equation to solve?. If it's an identity to prove, in general \(\displaystyle csc(x)-cos(x)\neq{sin(x)tan(x)}\)
If it's an equation:
\(\displaystyle csc(x)-cos(x)=sin(x)tan(x)\)
\(\displaystyle \frac{1}{sin(x)}-cos(x)=\frac{sin^{2}(x)}{cos(x)}\)
Multiply through by cos(x):
\(\displaystyle sin^{2}(x)=cot(x)-cos^{2}(x)\)
\(\displaystyle sin^{2}(x)+cos^{2}(x)=cot(x)\)
Now, can you finish?. Notice the left side, what that is?.
For the second one, try converting the csc[sup:2uo4fvvs]2[/sup:2uo4fvvs]t - 1 to cot(t) first and then convert everything in terms of sin and cos:
\(\displaystyle cost(csc^{2}t - 1) = cost \cdot cot^{2}t = cost \cdot \frac{cos^{2}t}{sin^{2}t} = \frac{cos^{3}t}{sin^{2}t}\)
Hmm. That's annoying. We have a power of 2 on the bottom and a power of 3 on the top. What can we multiply both top and bottom by :wink: ?
\(\displaystyle cost(csc^{2}t - 1) = cost \cdot cot^{2}t = cost \cdot \frac{cos^{2}t}{sin^{2}t} = \frac{cos^{3}t}{sin^{2}t}\)
Hmm. That's annoying. We have a power of 2 on the bottom and a power of 3 on the top. What can we multiply both top and bottom by :wink: ?
Is this the 3rd one?.
\(\displaystyle \frac{cot^{3}(t)}{csc(t)}=cos(t)(csc^{2}(t)-1)\)
If so,
\(\displaystyle \frac{cot^{3}(t)}{csc(t)}=\frac{\frac{cos^{3}(t)}{sin^{3}(t)}}{\frac{1}{sin(t)}}=\frac{cos^{3}(t)}{sin^{3}(t)}\cdot\frac{sin(t)}{1}=\frac{cos^{3}(t)}{sin^{2}(t)}=\frac{cos^{2}(t)}{sin^{2}(t)}\cdot\frac{cos(t)}{1}=cot^{2}(t)cos(t)\)\(\displaystyle =cos(t)(csc^{2}(t)-1)\)
\(\displaystyle \frac{cot^{3}(t)}{csc(t)}=cos(t)(csc^{2}(t)-1)\)
If so,
\(\displaystyle \frac{cot^{3}(t)}{csc(t)}=\frac{\frac{cos^{3}(t)}{sin^{3}(t)}}{\frac{1}{sin(t)}}=\frac{cos^{3}(t)}{sin^{3}(t)}\cdot\frac{sin(t)}{1}=\frac{cos^{3}(t)}{sin^{2}(t)}=\frac{cos^{2}(t)}{sin^{2}(t)}\cdot\frac{cos(t)}{1}=cot^{2}(t)cos(t)\)\(\displaystyle =cos(t)(csc^{2}(t)-1)\)