Question:
The value of
\[
\tan81^\circ-\tan63^\circ-\tan27^\circ+\tan9^\circ
\]
is
The value of
\[
\tan81^\circ-\tan63^\circ-\tan27^\circ+\tan9^\circ
\]
is
Show Hint
Convert angles like $81^\circ$ and $63^\circ$ into complementary angles first.
Updated On: Jun 3, 2026
- $2$
- $3$
- $4$
- $5$
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
Step 1: Concept
Use the identity \[ \tan(90^\circ-\theta)=\cot\theta. \]
Step 2: Meaning
Rewrite \[ \tan81^\circ=\cot9^\circ, \qquad \tan63^\circ=\cot27^\circ. \] Hence \[ E=\cot9^\circ-\cot27^\circ-\tan27^\circ+\tan9^\circ. \]
Step 3: Analysis
Using \[ \cot\theta-\tan\theta = 2\cot2\theta, \] we obtain \[ E=2\cot18^\circ-2\cot54^\circ. \] Applying standard exact trigonometric values gives \[ E=4. \]
Step 4: Conclusion
Therefore \[ \tan81^\circ-\tan63^\circ-\tan27^\circ+\tan9^\circ=4. \]
Final Answer: (C)
Use the identity \[ \tan(90^\circ-\theta)=\cot\theta. \]
Step 2: Meaning
Rewrite \[ \tan81^\circ=\cot9^\circ, \qquad \tan63^\circ=\cot27^\circ. \] Hence \[ E=\cot9^\circ-\cot27^\circ-\tan27^\circ+\tan9^\circ. \]
Step 3: Analysis
Using \[ \cot\theta-\tan\theta = 2\cot2\theta, \] we obtain \[ E=2\cot18^\circ-2\cot54^\circ. \] Applying standard exact trigonometric values gives \[ E=4. \]
Step 4: Conclusion
Therefore \[ \tan81^\circ-\tan63^\circ-\tan27^\circ+\tan9^\circ=4. \]
Final Answer: (C)
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