tan theta+1/tan theta=2 then prove that tan^theta+1/tan^theta=2
Answers
Answer:
Answer: The value of \bold{\tan ^{2} \theta+\frac{1}{\tan ^{2} \theta}=2}tan
2
θ+
tan
2
θ
1
=2
Given: \tan \theta+\frac{1}{\tan \theta}=2 \ldots(1)tanθ+
tanθ
1
=2…(1)
To find: \tan ^{2} \theta+\frac{1}{\tan ^{2} \theta}tan
2
θ+
tan
2
θ
1
Solution:
We know that by formula, (a+b)^{2}=a^{2}+b^{2}+2ab(a+b)
2
=a
2
+b
2
+2ab
Hence use the above formula to find \tan ^{2} \theta+\frac{1}{\tan ^{2} \theta}tan
2
θ+
tan
2
θ
1
\begin{gathered}\begin{array}{l}{\left(\tan \theta+\frac{1}{\tan \theta}\right)^{2}=\tan ^{2} \theta+\frac{1}{\tan ^{2} \theta}+2 \times \tan \theta \times \frac{1}{\tan \theta}} \\\\ {2^{2}=\tan ^{2} \theta+\frac{1}{\tan ^{2} \theta}+2 \times 1} \\ \\{4=\tan ^{2} \theta+\frac{1}{\tan ^{2} \theta}+2} \\\\ {\tan ^{2} \theta+\frac{1}{\tan ^{2} \theta}=4-2} \\\\ {\tan ^{2} \theta+\frac{1}{\tan ^{2} \theta}=2}\end{array}\end{gathered}
(tanθ+
tanθ
1
)
2
=tan
2
θ+
tan
2
θ
1
+2×tanθ×
tanθ
1
2
2
=tan
2
θ+
tan
2
θ
1
+2×1
4=tan
2
θ+
tan
2
θ
1
+2
tan
2
θ+
tan
2
θ
1
=4−2
tan
2
θ+
tan
2
θ
1
=2