Simplifying Trigonometric Expressions: Sin θ tan θ - sec θ

This question asks us to simplify a given trigonometric expression: $\sin \theta \tan \theta - \sec \theta$. To simplify this expression, we can use fundamental trigonometric identities to rewrite the terms in terms of $\sin \theta$ and $\cos \theta$.

Understanding the Trigonometric Functions

Let's break down the terms in the expression:

• $\sin \theta$: Sine of angle $\theta$.
• $\tan \theta$: Tangent of angle $\theta$. The identity for tangent is $\tan \theta = \frac{\sin \theta}{\cos \theta}$.
• $\sec \theta$: Secant of angle $\theta$. The identity for secant is $\sec \theta = \frac{1}{\cos \theta}$.

Step-by-Step Simplification of Sin θ tan θ - sec θ

Now, let's substitute the identities for $\tan \theta$ and $\sec \theta$ into the given expression:

The expression is: $\sin \theta \tan \theta - \sec \theta$

Substitute $\tan \theta = \frac{\sin \theta}{\cos \theta}$ and $\sec \theta = \frac{1}{\cos \theta}$:

$$ \sin \theta \left( \frac{\sin \theta}{\cos \theta} \right) - \frac{1}{\cos \theta} $$

Multiply the terms in the first part:

$$ \frac{\sin^2 \theta}{\cos \theta} - \frac{1}{\cos \theta} $$

Now we have two terms with a common denominator, $\cos \theta$. We can combine them:

$$ \frac{\sin^2 \theta - 1}{\cos \theta} $$

Recall the fundamental Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$.

We can rearrange this identity to find an expression for $\sin^2 \theta - 1$. Subtracting 1 from both sides gives:

$$ \sin^2 \theta + \cos^2 \theta - 1 = 1 - 1 $$

$$ \sin^2 \theta + \cos^2 \theta - 1 = 0 $$

Subtracting $\cos^2 \theta$ from both sides gives:

$$ \sin^2 \theta - 1 = -\cos^2 \theta $$

Now, substitute this into our simplified expression:

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

We can cancel one $\cos \theta$ term from the numerator and the denominator, assuming $\cos \theta \neq 0$:

$$ \frac{-\cos \theta \cdot \cos \theta}{\cos \theta} = -\cos \theta $$

So, the simplified expression is $-\cos \theta$.

Comparing with Options

Let's compare our result with the given options:

A. $-\cos \theta$

B. $1$

C. $-\sec \theta$

D. $\cosec \theta$

Our simplified expression $-\cos \theta$ matches option A.

The simplification process shows that $\sin \theta \tan \theta - \sec \theta$ simplifies to $-\cos \theta$.

Revision Table: Key Trigonometric Identities

Additional Information on Trigonometric Simplification

Simplifying trigonometric expressions is a common task in trigonometry. It often involves using fundamental identities to rewrite expressions in a simpler form, usually in terms of $\sin \theta$ and $\cos \theta$. Here are some tips:

• Convert to Sine and Cosine: A common strategy is to express all trigonometric functions (like $\tan$, $\cot$, $\sec$, $\csc$) in terms of $\sin \theta$ and $\cos \theta$.
• Look for Common Denominators: If you have fractions, combine them by finding a common denominator.
• Apply Pythagorean Identities: Be on the lookout for terms like $\sin^2 \theta + \cos^2 \theta$, $\sec^2 \theta - \tan^2 \theta$, or $\csc^2 \theta - \cot^2 \theta$, which simplify to 1. Also, rearrangements like $\sin^2 \theta - 1$ or $1 - \cos^2 \theta$ are useful.
• Factor and Cancel: After applying identities and combining terms, look for opportunities to factor expressions and cancel common factors in the numerator and denominator.
• Practice: Simplifying trigonometric expressions requires practice to become familiar with recognizing which identities to use.

Understanding these basic steps and identities is crucial for solving more complex trigonometric problems.