Simplifying Trigonometric Expressions: Step-by-Step Solution

We are asked to simplify the given trigonometric expression:

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

To simplify this expression, we need to combine the two fractions. First, find a common denominator, which is \( \cos\theta(1 + \cos\theta) \).

The first fraction already has the common denominator. For the second fraction, \( \frac{1 + \cos \theta}{\cos\theta} \), we need to multiply the numerator and the denominator by \( (1 + \cos\theta) \).

So the expression becomes:

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

This simplifies to:

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

Now that both fractions have the same denominator, we can combine their numerators:

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

Expand the term \( (1 + \cos \theta)^2 \) in the numerator:

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

Substitute this back into the numerator:

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

Rearrange the terms in the numerator to group \( \sin^2 \theta \) and \( \cos^2 \theta \):

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

Using the fundamental trigonometric identity \( \sin^2 \theta + \cos^2 \theta = 1 \), substitute 1 for \( (\sin^2 \theta + \cos^2 \theta) \):

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

Simplify the numerator:

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

Factor out 2 from the numerator:

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

Assuming \( 1 + \cos \theta \neq 0 \), we can cancel the term \( (1 + \cos \theta) \) from the numerator and the denominator:

$$ \frac{2}{\cos\theta} $$

Using the reciprocal identity \( \sec \theta = \frac{1}{\cos \theta} \), we can rewrite the expression as:

$$ 2 \times \frac{1}{\cos\theta} = 2\sec \theta $$

Thus, the simplified expression is \( 2\sec \theta \).

Revision Table: Key Trigonometric Identities

Additional Information on Trigonometric Functions

Trigonometric functions relate the angles of a right triangle to the ratios of its sides. The basic trigonometric functions are sine, cosine, and tangent. Their reciprocals are cosecant, secant, and cotangent.

• Sine (\( \sin \theta \)): Opposite side / Hypotenuse
• Cosine (\( \cos \theta \)): Adjacent side / Hypotenuse
• Tangent (\( \tan \theta \)): Opposite side / Adjacent side
• Cosecant (\( \csc \theta \)): \( \frac{1}{\sin \theta} \) = Hypotenuse / Opposite side
• Secant (\( \sec \theta \)): \( \frac{1}{\cos \theta} \) = Hypotenuse / Adjacent side
• Cotangent (\( \cot \theta \)): \( \frac{1}{\tan \theta} \) = Adjacent side / Opposite side

Simplifying trigonometric expressions often involves using these definitions and fundamental identities like the Pythagorean identities to rewrite the expression in a simpler form.