Question

If a straight line makes angles \( \alpha, \beta, \gamma \) with the coordinate axes, then evaluate \[ 1 - \frac{\tan^2\alpha}{1+\tan^2\alpha} + \frac{1}{\sec^2\beta} - 2\sin^2\gamma \]

• A. \( -1 \)
• B. \( 1 \)
• C. \( -2 \)
• D. \( 2 \)
• E. \( 0 \)

Hint:

Always reduce trig expressions using identities before substituting direction cosines.

Answer 1

Answer: (E)

Concept:
• Direction cosines: \[ l = \cos\alpha,\; m = \cos\beta,\; n = \cos\gamma \]
• Identity: \[ l^2 + m^2 + n^2 = 1 \]

Step 1: Simplify trigonometric expressions.
\[ \frac{\tan^2\alpha}{1+\tan^2\alpha} = \sin^2\alpha \] \[ \frac{1}{\sec^2\beta} = \cos^2\beta \]

Step 2: Rewrite expression.
\[ 1 - \sin^2\alpha + \cos^2\beta - 2\sin^2\gamma \] \[ = \cos^2\alpha + \cos^2\beta - 2\sin^2\gamma \]

Step 3: Use identity.
\[ \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1 \] So: \[ \cos^2\alpha + \cos^2\beta = 1 - \cos^2\gamma \]

Step 4: Substitute.
\[ = (1 - \cos^2\gamma) - 2\sin^2\gamma \]

Step 5: Convert everything to one function.
\[ \sin^2\gamma = 1 - \cos^2\gamma \] \[ = 1 - \cos^2\gamma - 2(1 - \cos^2\gamma) \] \[ = 1 - \cos^2\gamma - 2 + 2\cos^2\gamma \] \[ = -1 + \cos^2\gamma \]

Step 6: Use identity again.
\[ \cos^2\gamma = 1 - (\cos^2\alpha + \cos^2\beta) \] Substitute: \[ = -1 + 1 - (\cos^2\alpha + \cos^2\beta) \] \[ = -(\cos^2\alpha + \cos^2\beta) \] But: \[ \cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1 \] So final simplification gives: \[ 0 \]

Step 7: Final Answer.
\[ \boxed{0} \]