Show that the expression $\frac{\left(1-\cos ^2 x\right)\left(1+\cos ^2 x\right)}{\cos ^2 x}$ is equivalent to $\sin ^2 x+\sec ^2 x-1$.
Answer (2)
Use the identity sin 2 x + cos 2 x = 1 to rewrite the expression.
Simplify the expression to tan 2 x + sin 2 x .
Use the identity tan 2 x + 1 = sec 2 x to rewrite the expression.
Rearrange the terms to match the target expression: sin 2 x + sec 2 x − 1 .
Explanation
Understanding the Problem We are asked to show that the expression c o s 2 x ( 1 − c o s 2 x ) ( 1 + c o s 2 x ) is equivalent to sin 2 x + sec 2 x − 1 .
Using Trigonometric Identity We will start by simplifying the left-hand side of the equation. Recall the Pythagorean trigonometric identity: sin 2 x + cos 2 x = 1 . From this, we can write 1 − cos 2 x = sin 2 x .
Substituting the Identity Substituting this into the left-hand side, we get: cos 2 x ( sin 2 x ) ( 1 + cos 2 x ) .
Expanding the Numerator Now, we expand the numerator: cos 2 x sin 2 x + sin 2 x cos 2 x .
Splitting the Fraction Next, we split the fraction into two terms: cos 2 x sin 2 x + cos 2 x sin 2 x cos 2 x .
Simplifying the Expression We simplify each term. Recall that tan x = c o s x s i n x , so tan 2 x = c o s 2 x s i n 2 x . Also, the cos 2 x terms in the second fraction cancel out: tan 2 x + sin 2 x .
Using Another Trigonometric Identity Now, we use another trigonometric identity: tan 2 x + 1 = sec 2 x . From this, we can write tan 2 x = sec 2 x − 1 .
Substituting the Identity Substituting this into our expression, we get: sec 2 x − 1 + sin 2 x .
Rearranging the Terms Finally, we rearrange the terms to match the right-hand side of the original equation: sin 2 x + sec 2 x − 1 . Thus, we have shown that the two expressions are equivalent.
Conclusion Therefore, we have shown that \frac{\left(1-\cos ^2 x\right)\left(1+\cos ^2 x\){\cos ^2 x} is equivalent to sin 2 x + sec 2 x − 1 .
Examples
Trigonometric identities are useful in physics, especially when dealing with oscillatory motion, waves, and alternating current circuits. For example, when analyzing the motion of a pendulum or the behavior of an AC circuit, simplifying trigonometric expressions can make the equations easier to solve and interpret. This helps in predicting the behavior of these systems and designing them effectively.
The expression c o s 2 x ( 1 − c o s 2 x ) ( 1 + c o s 2 x ) simplifies to sin 2 x + sec 2 x − 1 using trigonometric identities. By substituting and rearranging terms, we confirm the equivalence. Hence, the two expressions are equal.
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