Math / Trigonometry

Question $1+\frac{1}{\tan ^{2} x}=\frac{1}{\sin ^{2} x}$

Studdy Solution

STEP 1

What is this asking? This is asking us to prove a trigonometric identity, showing that the left side equals the right side. Watch out! Remember your trigonometric identities and be careful not to mix them up!

STEP 2

1. Rewrite the tangent function
2. Simplify the left side
3. Use the Pythagorean identity

STEP 3

Alright, let's rewrite that tangent function in terms of sine and cosine!
We know that $\tan x = \frac{\sin x}{\cos x}$ , so $\tan^2 x = \frac{\sin^2 x}{\cos^2 x}$ .
This means $\frac{1}{\tan^2 x} = \frac{\cos^2 x}{\sin^2 x}$ .
Why are we doing this?
Because the right side of our identity has a $\sin^2 x$ term, so it's a good idea to get everything in terms of sines and cosines!

STEP 4

Substituting this back into the left side of our original equation gives us: $1 + \frac{\cos^2 x}{\sin^2 x}$

STEP 5

Now, let's simplify that left side.
To add the terms, we need a common denominator.
We can rewrite $1$ as $\frac{\sin^2 x}{\sin^2 x}$ .
This is just multiplying by one, so it doesn't change the value of our expression!
Now we have: $\frac{\sin^2 x}{\sin^2 x} + \frac{\cos^2 x}{\sin^2 x}$

STEP 6

With a common denominator, we can add the numerators: $\frac{\sin^2 x + \cos^2 x}{\sin^2 x}$

STEP 7

Here comes the magic!
Remember the Pythagorean identity: $\sin^2 x + \cos^2 x = 1$ .
Let's substitute that into our expression: $\frac{1}{\sin^2 x}$

STEP 8

BOOM! This is exactly the right side of our original identity.
We've proven it!

STEP 9

We've successfully shown that $1+\frac{1}{\tan ^{2} x}=\frac{1}{\sin ^{2} x}$ .

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