Question

প্রমাণ কর যে, $\frac{1}{1+tan^2A}+\frac{1}{1+cot^2A}=1$

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নবম-দশম শ্রেণির গণিত পাঠ্য বইয়ের ত্রিকোণমিতিক অনুপাত (৯.১) এর সকল সমাধান পেতে এখানে ক্লিক করুন।

Answer 1

Left Hand Side

$=\frac{1}{1+tan^2A}+\frac{1}{1+cot^2A}$

$=\frac{1}{1+tan^2A}+\left(\frac{1}{1+\frac{1}{tan^2A}}\right)$

$=\frac{1}{1+tan^2A}+\left(\frac{1}{\frac{tan^2A+1}{tan^2A}}\right)$

$=\frac{1}{1+tan^2A}+\left(1 \times \frac{tan^2A}{tan^2A+1}\right)$

$=\frac{1}{1+tan^2A}+\frac{tan^2A}{tan^2A+1}$

$=\frac{1}{1+tan^2A}+\frac{tan^2A}{1+tan^2A}$

$=\frac{1+tan^2A}{1+tan^2A}$

$=1$

$=$ Right Hand Side

[ Proved ]

Comments

Rule Applied :

• $cot \theta = \frac{1}{tan \theta}$

Answer 2

Left Hand Side

$=\frac{1}{1+tan^2A}+\frac{1}{1+cot^2A}$

$=\left(\frac{1}{1+\frac{sin^2A}{cos^2A}}\right)+\left(\frac{1}{1+\frac{cos^2A}{sin^2A}}\right)$

$=\left(\frac{1}{\frac{cos^2A+sin^2A}{cos^2A}}\right)+\left(\frac{1}{\frac{sin^2A+cos^2A}{sin^2A}}\right)$

$=\left(\frac{1}{\frac{1}{cos^2A}}\right)+\left(\frac{1}{\frac{1}{sin^2A}}\right)$

$=\left(1 \times \frac{cos^2A}{1}\right)+\left(1 \times \frac{sin^2A}{1}\right)$

$=cos^2A+sin^2A$

$=1$

$=$ Right Hand Side

[ Proved ]

Comments

Rules Applied :

• $tan \theta = \frac{sin \theta}{cos \theta}$
• $cot \theta = \frac{cos \theta}{sin \theta}$
• $sin^2\theta+cos^\theta=1$