Question

Which expression does not equal \(1 ?\) (a) \(\cos ^{2} \theta-\sin ^{2} \theta\) (b) \(\sec ^{2} \theta-\tan ^{2} \theta\) (c) \(\csc ^{2} \theta-\cot ^{2} \theta\) (d) \(\sin ^{2} \theta+\cos ^{2} \theta\)

Short answer

The expression \( \text{cos}^2 \theta - \text{sin}^2 \theta \) (Option a) does not equal 1.

Step-by-step solution

Evaluate Option (a)

The expression provided is \(\text{a) } \cos ^{2} \theta-\sin ^{2} \theta\). This is an identity known as the cosine of double angle: \(\text{cos}(2\theta)\). This is not equal to 1 for all \( \theta \).

Evaluate Option (b)

The expression provided is \(\text{b) } \sec ^{2} \theta - \tan ^{2} \theta\). Remember that \(\text{sec}(\theta) = \frac{1}{\text{cos}(\theta)}\) and \(\text{tan}(\theta) = \frac{\text{sin}(\theta)}{\text{cos}(\theta)}\). Using the Pythagorean identity \(\text{sec}^2 \theta = 1 + \text{tan}^2 \theta\), we get: \(\text{sec}^2 \theta - \text{tan}^2 \theta = 1\).

Evaluate Option (c)

The expression provided is \(\text{c) } \csc ^{2} \theta - \cot ^{2} \theta\). Recall that \(\text{csc}(\theta) = \frac{1}{\text{sin}(\theta)} \) and \(\text{cot}(\theta) = \frac{\text{cos}(\theta)}{\text{sin}(\theta)}\). Using the Pythagorean identity \(\text{csc}^2 \theta = 1 + \text{cot}^2 \theta\), we get: \(\text{csc}^2 \theta - \text{cot}^2 \theta = 1\).

Evaluate Option (d)

The expression provided is \(\text{d) } \sin ^{2} \theta + \cos ^{2} \theta\). According to the fundamental Pythagorean identity, we know that \(\text{sin}^2 \theta + \text{cos}^2 \theta = 1\).

Identify the Expression that does not equal 1

Based on the evaluations: \( \text{cos}^2 \theta - \text{sin}^2 \theta \) (Option a) is not equal to 1 for all \( \theta \), while all other options equal 1.

Key concepts

cosine double angle identity

The cosine double angle identity is a way to express the cosine of twice an angle, or \(2\theta\), in different forms. The identity can be written as \[\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)\]. This simplifies the process of working with trigonometric functions involving double angles.
It's important to note that \(\cos^2(\theta) - \sin^2(\theta)\) is equal to \(\cos(2\theta)\) and does not always equal 1 for all \(\theta\).
This was the critical insight needed to determine that expression (a) \(\cos^2(\theta) - \sin^2(\theta)\) in the exercise does not equal 1.

Pythagorean identity

The Pythagorean identity is one of the fundamental relationships in trigonometry. It states that \[\sin^2(\theta) + \cos^2(\theta) = 1\]. This identity is derived from the Pythagorean theorem and applies to any angle \(\theta\).
This identity allows us to express one trigonometric function in terms of another. For example, if we know \(\sin(\theta)\), we can find \(\cos(\theta)\) using the equation \[\cos(\theta) = \sqrt{1 - \sin^2(\theta)}\].
In the exercise, option (d) \(\sin^2(\theta) + \cos^2(\theta) \) clearly equals 1 due to this identity.

secant and tangent identity

The secant and tangent identity is an extension of the Pythagorean identity. It states that \[\sec^2(\theta) = 1 + \tan^2(\theta)\].
To understand this, recall the definitions of secant and tangent:

• \(\sec(\theta) = \frac{1}{\cos(\theta)}\)
• \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)

Substituting these into the Pythagorean identity, we get the secant and tangent relationship.
This identity was used to evaluate option (b) \(\sec^2(\theta) - \tan^2(\theta)\), showing that it equals 1.

cosecant and cotangent identity

Similar to the previous identities, the relationship between cosecant and cotangent derives from the Pythagorean identity. The identity is \[\csc^2(\theta) = 1 + \cot^2(\theta)\].
Here, we use the definitions:

• \(\csc(\theta) = \frac{1}{\sin(\theta)}\)
• \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\)

By substituting these into the Pythagorean identity, we can verify the cosecant and cotangent relationship.
This identity was instrumental in confirming that option (c) \(\csc^2(\theta) - \cot^2(\theta)\) equals 1.