Question

Analytically show that the equation below represents trigonometric identity statements for questions 5-7

Answer 1

Analytically show that the equation below represents trigonometric

Answer 2

The analytical derivations confirm that Equation 5, Equation 7, and Equation 6 represent valid trigonometric identity statements, as the left-hand sides simplify to match their respective right-hand sides.

The given trigonometric equations and show analytically that they represent trigonometric identity statements.

Equation 5:

`\[ \sin(x) \csc(x) + (\csc(x) + \sin(x) \sec^2(x)) = \sec^2(x) \]`

We'll start by simplifying the left-hand side (LHS):

`\[ \text{LHS} = \sin(x) \csc(x) + \csc(x) + \sin(x) \sec^2(x) \]`

Now, rewrite csc(x) as
`\( (1)/(\sin(x)) \)` and
`\(\sec^2(x)\)` as
`\( (1)/(\cos^2(x)) \)`:

`\[ \text{LHS} = (\sin(x))/(\sin(x)) + (1)/(\sin(x)) + (\sin(x))/(\cos^2(x)) \]`

Combine the fractions:

`\[ \text{LHS} = (\sin(x) + 1 + \sin(x))/(\sin(x)) \cdot (\cos^2(x))/(\cos^2(x)) \]`

Combine like terms:

`\[ \text{LHS} = (2\sin(x) + 1)/(\sin(x)) \cdot (\cos^2(x))/(\cos^2(x)) \]`

`\[ \text{LHS} = (2\sin(x) + 1)/(\sin(x)) \]`

Now, multiply the numerator and denominator by
`\(\cos^2(x)\)`:

`\[ \text{LHS} = ((2\sin(x) + 1)\cos^2(x))/(\sin(x)\cos^2(x)) \]`

Apply the trigonometric identity
`\( \sin(x)\cos^2(x) = \sin(x) \)`:

`\[ \text{LHS} = (2\sin(x) + 1)/(\sin(x)) \]`

`\[ \text{LHS} = 2 + (1)/(\sin(x)) \]`

Now, rewrite
`\( (1)/(\sin(x)) \)` as csc(x):

`\[ \text{LHS} = 2 + \csc(x) \]`

Now, compare the simplified LHS with the given RHS
`(\( \sec^2(x) \))`:

`\[ 2 + \csc(x) = \sec^2(x) \]`

Hence, Equation 5 is satisfied.

Equation 7:

`\[ (1 + \sec(\theta))/(\tan(\theta)) + (\tan(\theta))/(1 + \sec(\theta)) = 2\csc(\theta) \]`

Start by finding a common denominator for the two fractions on the LHS, which is
`\( \tan(\theta)(1 + \sec(\theta)) \)`:

`\[ \text{LHS} = ((1 + \sec(\theta))^2 + \tan^2(\theta))/(\tan(\theta)(1 + \sec(\theta))) \]`

Now, expand and simplify:

`\[ \text{LHS} = (1 + 2\sec(\theta) + \sec^2(\theta) + \tan^2(\theta))/(\tan(\theta)(1 + \sec(\theta))) \]`

Apply the trigonometric identity
`\( \sec^2(\theta) = \tan^2(\theta) + 1 \)`:

`\[ \text{LHS} = (2 + 2\sec(\theta))/(\tan(\theta)(1 + \sec(\theta))) \]`

Combine like terms:

`\[ \text{LHS} = (2(1 + \sec(\theta)))/(\tan(\theta)(1 + \sec(\theta))) \]`

Now, cancel out common factors:

`\[ \text{LHS} = (2)/(\tan(\theta)) \]`

Recall that
`\( \tan(\theta) = (1)/(\cot(\theta)) \)`, so substitute this in:

`\[ \text{LHS} = 2\cot(\theta) \]`

Finally, use the identity
`\( \cot(\theta) = (1)/(\tan(\theta)) \)`:

`\[ \text{LHS} = 2\cot(\theta) = 2\csc(\theta) \]`

Therefore, Equation 7 is satisfied.

Equation 6:

`\[ \sec^2(\theta) + \csc^2(\theta) = \sec^2(\theta)\csc^2(\theta) \]`

We'll start by using the definitions of
`\( \sec^2(\theta) \) and \( \csc^2(\theta) \)`:

`\[ \text{LHS} = (1)/(\cos^2(\theta)) + (1)/(\sin^2(\theta)) \]`

Combine the fractions:

`\[ \text{LHS} = (\sin^2(\theta) + \cos^2(\theta))/(\sin^2(\theta)\cos^2(\theta)) \]`

Apply the Pythagorean identity
`\( \sin^2(\theta) + \cos^2(\theta) = 1 \)`:

`\[ \text{LHS} = (1)/(\sin^2(\theta)\cos^2(\theta)) \]`

`\[ \text{LHS} = \sec^2(\theta)\csc^2(\theta) \]`

Therefore, Equation 6 is satisfied.

All three given equations represent trigonometric identity statements.