Prove Trigonometric Identities - Pre-Calculus

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Question

Simplify: $1+\frac{cos^2(\theta)}{sin^2(\theta)}$

Answer

To simplify $1+\frac{cos^2(\theta)}{sin^2(\theta)}$ , find the common denominator and multiply the numerator accordingly.

$1+\frac{cos^2(\theta)}{sin^2(\theta)}=\frac{sin^2(\theta)}{sin^2(\theta)}+\frac{cos^2(\theta)}{sin^2(\theta)}=\frac{sin^2(\theta)+cos^2(\theta)}{sin^2(\theta)}$

The numerator is an identity.

$sin^2(\theta)+cos^2(\theta)=1$

Substitute the identity and simplify.

$\frac{sin^2(\theta)+cos^2(\theta)}{sin^2(\theta)}= \frac{1}{sin^2(\theta)}= csc^2(\theta)$

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Question

Simplify the following:

$sin^3(x)+sin(x)\cdot cos^2(x)$

Answer

$sin^3(x)+sin(x)\cdot cos^2(x)$

First factor out sine x.

$=(sin^2(x)+cos^2(x))\cdot sin(x)$

Notice that a Pythagorean Identity is present.

The identity needed for this problem is:

Using this identity the equation becomes,

$=(1)\cdot sin(x)$

.

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Question

Evaluate in terms of sines and cosines:

$\frac{sec^2(\theta)}{cot(\theta)}$

Answer

Convert $\frac{sec^2(\theta)}{cot(\theta)}$ into its sines and cosines.

$\frac{sec^2(\theta)}{cot(\theta)}= \left(\frac{1}{cos(\theta)}\times\frac{1}{cos(\theta)}\right)\div \frac{cos(\theta)}{sin(\theta)}$

$\left(\frac{1}{cos(\theta)}\times\frac{1}{cos(\theta)}\right)\div \frac{cos(\theta)}{sin(\theta)}=\frac{1}{cos^2(\theta)}\times\frac{sin(\theta)}{cos(\theta)}= \frac{sin(\theta)}{cos^3(\theta)}$

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Question

Simplify the expression $\frac{sin(x)}{cos(x)}\times \frac{1}{cot(x)}$

Answer

To simplify, use the trigonometric identities $\frac{sin(x)}{cos(x)}=tan(x)$ and $\frac{1}{cot(x)}=tan(x)$ to rewrite both halves of the expression:

$tan(x)\times tan(x)$

Then combine using an exponent to simplify:

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Question

Simplify $\frac{cos(x)}{sin(x)}$ .

Answer

This expression is a trigonometric identity: $\frac{cos(x)}{sin(x)}=cot(x)$

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Question

Simplify $\frac{2}{csc^2(x)}+\frac{2}{sec^2(x)}$

Answer

Factor out 2 from the expression:

$2(\frac{1}{csc^2(x)}+\frac{1}{sec^2(x)})$

Then use the trigonometric identities $\frac{1}{csc(x)}=sin(x)$ and $\frac{1}{sec(x)}=cos(x)$ to rewrite the fractions:

Finally, use the trigonometric identity to simplify:

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Question

Simplify $\frac{2sin^2(x)}{tan(x)}+\frac{2cos^2(x)}{tan(x)}$

Answer

Factor out the common $\frac{2}{tan(x)}$ from the expression:

$(\frac{2}{tan(x)})(sin^2(x)+cos^2(x))$

Next, use the trigonometric identify to simplify:

$\frac{2}{tan(x)}(1)$

Then use the identify $\frac{1}{tan(x)}=cot(x)$ to simplify further:

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Question

Simplify $\frac{2tan^2x+1}{tan^2x}+tan^2x$

Answer

To simplify the expression, separate the fraction into two parts:

$\frac{2tan^2x}{tan^2x}+\frac{1}{tan^2x}+tan^2x$

The terms in the first fraction cancel leaving you with:

$2+\frac{1}{tan^2x}+tan^2x$

Then you can deal with the remaining fraction using the rule that $\frac{1}{tan^2x}=cot^2x$ . This leaves:

You can separate this into:

And each half of this expression is now a trigonometric identity: and . This gives you:

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