Evaluating Trigonometric Expressions using Complementary Angles

The question asks us to find the value of the expression $2\tan55^\circ + \cot55^\circ$ given that $Y = \tan35^\circ$. To solve this, we need to express $\tan55^\circ$ and $\cot55^\circ$ in terms of $Y$, which is $\tan35^\circ$. We can use the properties of complementary angles.

Understanding Complementary Angles in Trigonometry

Two angles are complementary if their sum is $90^\circ$. In this problem, we notice that $55^\circ + 35^\circ = 90^\circ$. This means $55^\circ$ and $35^\circ$ are complementary angles. The key trigonometric identities involving complementary angles are:

• $\tan(90^\circ - \theta) = \cot\theta$
• $\cot(90^\circ - \theta) = \tan\theta$
• $\cot\theta = \frac{1}{\tan\theta}$

Relating tan55° and cot55° to tan35°

Since $55^\circ = 90^\circ - 35^\circ$, we can apply the identities:

• $\tan55^\circ = \tan(90^\circ - 35^\circ)$
• Using the identity $\tan(90^\circ - \theta) = \cot\theta$, we get $\tan55^\circ = \cot35^\circ$.

We know that $\cot\theta = \frac{1}{\tan\theta}$. Therefore, $\cot35^\circ = \frac{1}{\tan35^\circ}$.

Given that $Y = \tan35^\circ$, we have:

• $\tan55^\circ = \cot35^\circ = \frac{1}{\tan35^\circ} = \frac{1}{Y}$

Similarly, for $\cot55^\circ$:

• $\cot55^\circ = \cot(90^\circ - 35^\circ)$
• Using the identity $\cot(90^\circ - \theta) = \tan\theta$, we get $\cot55^\circ = \tan35^\circ$.

Since $Y = \tan35^\circ$, we have:

• $\cot55^\circ = \tan35^\circ = Y$

Substituting into the Expression

Now we substitute the values we found for $\tan55^\circ$ and $\cot55^\circ$ in terms of $Y$ into the given expression $(2\tan55^\circ + \cot55^\circ)$.

The expression is $2\tan55^\circ + \cot55^\circ$.

Substitute $\tan55^\circ = \frac{1}{Y}$ and $\cot55^\circ = Y$:

$2\left(\frac{1}{Y}\right) + Y$

Simplify the expression:

$\frac{2}{Y} + Y$

To combine these terms, find a common denominator, which is $Y$:

$\frac{2}{Y} + \frac{Y \cdot Y}{Y}$

$\frac{2}{Y} + \frac{Y^2}{Y}$

Combine the numerators over the common denominator:

$\frac{2 + Y^2}{Y}$

Comparing with the Options

The simplified expression is $\frac{2 + Y^2}{Y}$. Let's compare this with the given options:

• Option 1: $\frac{2}{Y^2}$
• Option 2: $\frac{2 - Y^2}{Y}$
• Option 3: $\frac{2 - Y}{Y^2}$
• Option 4: $\frac{2 + Y^2}{Y}$

Our result matches Option 4.

Conclusion

By using the complementary angle relationships between $55^\circ$ and $35^\circ$ and the definition of $Y = \tan35^\circ$, we successfully transformed the expression $2\tan55^\circ + \cot55^\circ$ into a form expressed in terms of $Y$. The resulting expression is $\frac{2 + Y^2}{Y}$.

Revision Table: Trigonometry Concepts

Additional Information: Working with Trigonometric Expressions

When dealing with trigonometric expressions involving different angles, look for relationships between the angles. Complementary angles ($90^\circ - \theta$) and supplementary angles ($180^\circ - \theta$) are common relationships. Understanding the fundamental identities (reciprocal identities, quotient identities, Pythagorean identities) is crucial for simplifying expressions and solving equations.

• Reciprocal Identities: $\sin\theta = \frac{1}{\csc\theta}$, $\cos\theta = \frac{1}{\sec\theta}$, $\tan\theta = \frac{1}{\cot\theta}$.
• Quotient Identities: $\tan\theta = \frac{\sin\theta}{\cos\theta}$, $\cot\theta = \frac{\cos\theta}{\sin\theta}$.
• Pythagorean Identities: $\sin^2\theta + \cos^2\theta = 1$, $1 + \tan^2\theta = \sec^2\theta$, $1 + \cot^2\theta = \csc^2\theta$.

In this problem, recognizing the complementary nature of $55^\circ$ and $35^\circ$ was the key first step. This allowed us to express the terms in the expression using the given variable $Y = \tan35^\circ$.