Proving Trig Identities

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Pre-Calculus › Proving Trig Identities

Questions 1 - 10

Simplify the following:

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

The expression is already in simplified form

Explanation

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

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,

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

Simplify the following:

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

The expression is already in simplified form

Explanation

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

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,

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

Evaluate in terms of sines and cosines:

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

$\frac{sin(\theta)}{cos^3(\theta)}$

$\frac{sin(\theta)}{cos(\theta)}$

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

$\frac{1}{cos(\theta)sin(\theta)}$

$\frac{cos(\theta)}{sin(\theta)}$

Explanation

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

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

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

Evaluate in terms of sines and cosines:

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

$\frac{sin(\theta)}{cos^3(\theta)}$

$\frac{sin(\theta)}{cos(\theta)}$

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

$\frac{1}{cos(\theta)sin(\theta)}$

$\frac{cos(\theta)}{sin(\theta)}$

Explanation

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

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

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

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