Use identities to simplify the expression.
[tex]\tan ^2 \theta-\frac{1}{\cos ^2 \theta}[/tex]
Answer (1)
Rewrite tan 2 θ as c o s 2 θ s i n 2 θ .
Combine the fractions to get c o s 2 θ s i n 2 θ − 1 .
Use the Pythagorean identity sin 2 θ − 1 = − cos 2 θ .
Simplify the expression to get − 1 .
Explanation
Understanding the Problem We are asked to simplify the expression tan 2 θ − c o s 2 θ 1 using trigonometric identities.
Rewriting the Expression We know that tan θ = c o s θ s i n θ , so tan 2 θ = c o s 2 θ s i n 2 θ . Substituting this into the expression, we get: cos 2 θ sin 2 θ − cos 2 θ 1
Combining Fractions Since both terms have a common denominator, we can combine them: cos 2 θ sin 2 θ − 1
Using the Pythagorean Identity We know the Pythagorean identity sin 2 θ + cos 2 θ = 1 , which can be rearranged to sin 2 θ − 1 = − cos 2 θ . Substituting this into the expression, we have: cos 2 θ − cos 2 θ
Simplifying the Expression Finally, we simplify the expression: cos 2 θ − cos 2 θ = − 1
Final Answer Therefore, the simplified expression is − 1 .
Examples
Trigonometric identities are useful in various fields such as physics and engineering. For example, when analyzing the motion of a pendulum, simplifying trigonometric expressions can help in determining the period of oscillation. Similarly, in electrical engineering, simplifying expressions involving trigonometric functions can aid in analyzing alternating current circuits. These identities provide a way to rewrite complex expressions into simpler forms, making calculations and analysis more manageable.
