Key Concepts and Formulas

• Direction Cosines of a Line: For a line in 3D space, if it makes angles $\alpha$, $\beta$, and $\gamma$ with the positive $x$, $y$, and $z$-axes respectively, then $\cos\alpha$, $\cos\beta$, and $\cos\gamma$ are called its direction cosines. They are often denoted as $l$, $m$, and $n$.
• Fundamental Identity of Direction Cosines: The sum of the squares of the direction cosines of any line is always equal to 1. ${\cos ^2}\alpha + {\cos ^2}\beta + {\cos ^2}\gamma = 1$
• Pythagorean Trigonometric Identity: For any angle $x$, the following identity holds: ${\sin ^2}x + {\cos ^2}x = 1$ This can be rearranged as ${\cos ^2}x = 1 - {\sin ^2}x$ or ${\sin ^2}x = 1 - {\cos ^2}x$.

Step-by-Step Solution

Step 1: Identify the angles the line makes with the coordinate axes. The problem states that the line makes an angle $\theta$ with the $x$-axis, an angle $\beta$ with the $y$-axis, and an angle $\theta$ with the $z$-axis.

• Why this step: This is the initial interpretation of the given information, mapping the problem's description to the standard notation for direction cosines. Therefore, we have:
• Angle with $x$-axis, $\alpha = \theta$
• Angle with $y$-axis, $\beta = \beta$
• Angle with $z$-axis, $\gamma = \theta$

Step 2: Apply the fundamental identity of direction cosines. Substitute the identified angles into the fundamental identity ${\cos ^2}\alpha + {\cos ^2}\beta + {\cos ^2}\gamma = 1$.

• Why this step: This identity is the core geometric principle that connects the angles a line makes with the coordinate axes. It forms the primary equation for solving the problem. Substituting the values, we get: ${\cos ^2}\theta + {\cos ^2}\beta + {\cos ^2}\theta = 1$

Step 3: Simplify the equation. Combine the like terms involving ${\cos ^2}\theta$.

• Why this step: Simplifying the equation makes it easier to manipulate algebraically. $2{\cos ^2}\theta + {\cos ^2}\beta = 1$

Step 4: Use the Pythagorean trigonometric identity to relate $\cos^2\beta$ and $\sin^2\beta$. The problem provides a condition involving ${\sin ^2}\beta$. To use this condition, we need to express ${\cos ^2}\beta$ in terms of ${\sin ^2}\beta$ using the identity ${\sin ^2}x + {\cos ^2}x = 1$.

• Why this step: This transformation is crucial for connecting our derived equation with the specific condition given in the problem statement, which involves ${\sin ^2}\beta$. From the identity, we know ${\cos ^2}\beta = 1 - {\sin ^2}\beta$. Substitute this into the equation from Step 3: $2{\cos ^2}\theta + (1 - {\sin ^2}\beta) = 1$ Rearrange the terms to isolate ${\sin ^2}\beta$: $2{\cos ^2}\theta - {\sin ^2}\beta = 1 - 1$ $2{\cos ^2}\theta = {\sin ^2}\beta$

Step 5: Substitute the given condition into the equation. The problem states the relationship: ${\sin ^2}\beta = 3{\sin ^2}\theta$.

• Why this step: This is a critical substitution that eliminates the angle $\beta$ from our equation, leaving an equation solely in terms of $\theta$, which is what we need to solve for. Substitute this into the equation from Step 4: $2{\cos ^2}\theta = 3{\sin ^2}\theta$

Step 6: Convert $\sin^2\theta$ to $\cos^2\theta$ and solve. To find the value of ${\cos ^2}\theta$, we need to express the entire equation in terms of ${\cos ^2}\theta$. We use the Pythagorean identity again: ${\sin ^2}\theta = 1 - {\cos ^2}\theta$.

• Why this step: This allows us to have a single variable (${\cos ^2}\theta$) in the equation, making it solvable algebraically. Substitute this into the equation from Step 5: $2{\cos ^2}\theta = 3(1 - {\cos ^2}\theta)$

Step 7: Expand and solve for $\cos^2\theta$. Distribute the 3 on the right side and collect all terms involving ${\cos ^2}\theta$.

• Why this step: These are standard algebraic steps to isolate the desired variable and find its value. $2{\cos ^2}\theta = 3 - 3{\cos ^2}\theta$ Add $3{\cos ^2}\theta$ to both sides: $2{\cos ^2}\theta + 3{\cos ^2}\theta = 3$ $5{\cos ^2}\theta = 3$ Divide by 5: ${\cos ^2}\theta = {3 \over 5}$

Common Mistakes & Tips

• Forgetting the fundamental identity: The relation ${\cos ^2}\alpha + {\cos ^2}\beta + {\cos ^2}\gamma = 1$ is the cornerstone of 3D geometry problems involving angles a line makes with the axes. Always start by writing it down.
• Errors in trigonometric identities: Be careful when converting between sine and cosine squares (e.g., $1 - {\cos ^2}x = {\sin ^2}x$).
• Algebraic blunders: Pay close attention to distribution, combining like terms, and rearranging equations. Simple calculation errors can lead to an incorrect final answer.
• Reading the question: Ensure you are solving for the quantity asked (here, ${\cos ^2}\theta$, not $\cos\theta$ or $\theta$).

Summary

This problem effectively combines the geometric concept of direction cosines with fundamental trigonometric identities. By first applying the direction cosine identity, we established a relationship between the angles. Then, using the Pythagorean identity and the given condition, we systematically transformed the equation to express it solely in terms of ${\cos ^2}\theta$. Finally, algebraic manipulation yielded the required value.

The final answer is $\boxed{{3 \over 5}}$, which corresponds to option (C).