Finding the False Trigonometry Statement

The question asks us to identify the statement that is FALSE among the given trigonometric identities. To do this, we will examine each statement and check if it holds true based on fundamental trigonometric identities.

Recalling Key Trigonometric Identities

We will use the following well-known trigonometric identities:

• Pythagorean Identity: $\sin^2 A + \cos^2 A = 1$
• Derived Pythagorean Identities:
  • $1 + \tan^2 A = \sec^2 A$
  • $1 + \cot^2 A = \cosec^2 A$
• Reciprocal Identities:
  • $\sec A = \frac{1}{\cos A}$
  • $\cosec A = \frac{1}{\sin A}$
  • $\cot A = \frac{1}{\tan A}$
• Ratio Identities:
  • $\tan A = \frac{\sin A}{\cos A}$
  • $\cot A = \frac{\cos A}{\sin A}$

Analyzing Each Trigonometric Statement

Let's check each given statement one by one:

Statement 1: $\tan^2 A = 1 - \sec^2 A$

We know the Pythagorean identity $1 + \tan^2 A = \sec^2 A$. Let's rearrange this identity to express $\tan^2 A$:

From $1 + \tan^2 A = \sec^2 A$, we subtract 1 from both sides:

$\tan^2 A = \sec^2 A - 1$

Now let's look at the given statement: $\tan^2 A = 1 - \sec^2 A$. We can rewrite $1 - \sec^2 A$ as $-(\sec^2 A - 1)$.

So the statement is $\tan^2 A = -(\sec^2 A - 1)$.

Comparing this with the correct identity $\tan^2 A = \sec^2 A - 1$, we see that the given statement claims $\tan^2 A = - \tan^2 A$. This is only true if $\tan^2 A = 0$, which means $\tan A = 0$. This identity does not hold true for all values of A where $\tan A$ and $\sec A$ are defined. Therefore, this statement is FALSE.

Statement 2: $\sin A = \tan A \times \cos A$

We know the ratio identity $\tan A = \frac{\sin A}{\cos A}$ (assuming $\cos A \neq 0$). Let's substitute this into the right side of the statement:

Right Side = $\tan A \times \cos A = \left(\frac{\sin A}{\cos A}\right) \times \cos A$

Assuming $\cos A \neq 0$, we can cancel $\cos A$:

Right Side = $\sin A$

The statement becomes $\sin A = \sin A$, which is TRUE for all values of A where $\tan A$ and $\cos A$ are defined and $\cos A \neq 0$.

Statement 3: $\cosec^2 A - 1 = \cot^2 A$

We know the Pythagorean identity $1 + \cot^2 A = \cosec^2 A$. Let's rearrange this identity to isolate $\cot^2 A$:

From $1 + \cot^2 A = \cosec^2 A$, we subtract 1 from both sides:

$\cot^2 A = \cosec^2 A - 1$

This matches the given statement exactly. Therefore, this statement is TRUE for all values of A where $\cot A$ and $\cosec A$ are defined.

Statement 4: $\cos A \times \sec A = 1$

We know the reciprocal identity $\sec A = \frac{1}{\cos A}$ (assuming $\cos A \neq 0$). Let's substitute this into the left side of the statement:

Left Side = $\cos A \times \sec A = \cos A \times \left(\frac{1}{\cos A}\right)$

Assuming $\cos A \neq 0$, we can cancel $\cos A$:

Left Side = 1

The statement becomes $1 = 1$, which is TRUE for all values of A where $\cos A$ and $\sec A$ are defined and $\cos A \neq 0$.

Conclusion: Identifying the False Trigonometry Statement

Based on our analysis of each trigonometric statement, we found that statement 1 is the only one that is FALSE. The correct identity relating $\tan^2 A$ and $\sec^2 A$ is $\tan^2 A = \sec^2 A - 1$, not $\tan^2 A = 1 - \sec^2 A$.

Revision Table: Fundamental Trigonometric Identities

Additional Information on Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variable for which the expressions are defined. They are crucial in simplifying trigonometric expressions, solving trigonometric equations, and calculating derivatives and integrals of trigonometric functions.

• The domain of validity for trigonometric identities depends on the specific functions involved. For example, identities involving $\tan A$ or $\sec A$ are not defined when $\cos A = 0$ (i.e., $A = \frac{\pi}{2} + n\pi$ for integer $n$). Identities involving $\cot A$ or $\cosec A$ are not defined when $\sin A = 0$ (i.e., $A = n\pi$ for integer $n$).
• Understanding the relationship between the six trigonometric functions (sin, cos, tan, cot, sec, cosec) is key to mastering identities. They are interconnected through reciprocal, ratio, and Pythagorean relationships.
• Practicing with different types of problems, such as simplifying expressions or proving identities, helps reinforce the understanding and application of these fundamental trigonometric identities.