Question
Second square theta minus 10 square theta is equal to 1 true or false
Answers
GPT-4o mini
To determine whether the equation \( 2 \sec^2 \theta - 10 \tan^2 \theta = 1 \) is true or false, we can manipulate the equation using trigonometric identities.
Recall the following trigonometric identities:
1. \( \sec^2 \theta = 1 + \tan^2 \theta \)
Substituting \( \sec^2 \theta \) into the equation:
\[ 2(1 + \tan^2 \theta) - 10 \tan^2 \theta = 1 \]
Expanding this gives:
\[ 2 + 2\tan^2 \theta - 10\tan^2 \theta = 1 \]
Combining like terms:
\[ 2 - 8\tan^2 \theta = 1 \]
Now, isolate the term involving \( \tan^2 \theta \):
\[ -8\tan^2 \theta = 1 - 2 \]
\[ -8\tan^2 \theta = -1 \]
Dividing both sides by -8:
\[ \tan^2 \theta = \frac{1}{8} \]
This implies that there are angles \( \theta \) that satisfy this equation. Therefore, the original equation can be true for specific values of \( \theta \).
So, the conclusion is that the statement is conditionally true; there exist values of \( \theta \) that satisfy \( 2 \sec^2 \theta - 10 \tan^2 \theta = 1 \).
Recall the following trigonometric identities:
1. \( \sec^2 \theta = 1 + \tan^2 \theta \)
Substituting \( \sec^2 \theta \) into the equation:
\[ 2(1 + \tan^2 \theta) - 10 \tan^2 \theta = 1 \]
Expanding this gives:
\[ 2 + 2\tan^2 \theta - 10\tan^2 \theta = 1 \]
Combining like terms:
\[ 2 - 8\tan^2 \theta = 1 \]
Now, isolate the term involving \( \tan^2 \theta \):
\[ -8\tan^2 \theta = 1 - 2 \]
\[ -8\tan^2 \theta = -1 \]
Dividing both sides by -8:
\[ \tan^2 \theta = \frac{1}{8} \]
This implies that there are angles \( \theta \) that satisfy this equation. Therefore, the original equation can be true for specific values of \( \theta \).
So, the conclusion is that the statement is conditionally true; there exist values of \( \theta \) that satisfy \( 2 \sec^2 \theta - 10 \tan^2 \theta = 1 \).