Question

Tan²5.tan²10.tan²15.tan²20........tan²90=???

Answer 1

${ \tan }^{2}5 \times { \tan }^{2} 85 \times { \tan }^{2}10 \times { \tan }^{2}80 \times { \tan }^{2}15 \times { \tan }^{2}75 \times { \tan }^{2}20 \times { \tan }^{2}70 \times { \tan }^{2}25 \times { \tan }^{2}65 \times { \tan }^{2}30 \times { \tan }^{2}60 \times { \tan }^{2}35 \times { \tan }^{2}55 \times { \tan }^{2}40 \times { \tan }^{2}50 \times { \tan }^{2}45 \times { \tan }^{2}90 \\ = { \tan }^{2}5 \times { \cot }^{2} 5 \times { \tan }^{2}10 \times { \cot }^{2}10 \times { \tan }^{2}15 \times { \cot }^{2}15 \times { \tan }^{2}20 \times { \cot}^{2}20 \times { \tan }^{2}25 \times { \cot}^{2}25 \times { \tan }^{2}30 \times { \cot}^{2}30 \times { \tan }^{2}35 \times { \cot }^{2}35 \times { \tan }^{2}40 \times { \cot }^{2}40 \times { \tan }^{2}45 \times { \tan }^{2}90 \\ = { \tan }^{2}5 \times \frac{1}{{ \tan }^{2}5} \times { \tan }^{2}10 \times \frac{1}{{ \tan }^{2}10} \times { \tan }^{2}15 \times \frac{1}{{ \tan }^{2}15} \times { \tan }^{2}20 \times \frac{1}{{ \tan }^{2}20} \times { \tan }^{2}25 \times \frac{1}{{ \tan }^{2}25} \times { \tan }^{2}30 \times \frac{1}{{ \tan }^{2}30} \times { \tan }^{2}35 \times \frac{1}{{ \tan }^{2}35} \times { \tan }^{2}40 \times \frac{1}{{ \tan }^{2}40} \times { \tan }^{2}45 \times { \tan }^{2}90 \\ = 1 \times 1 \times 1 \times 1 \times 1 \times 1 \times 1 \times 1 \times { \tan }^{2}45 \times { \tan }^{2}90 \\ = 1 \times {1}^{2} \times 1 \\ = 1$