Tan -1 x-cot -1x- -2-x- R - Maths - Q-A

Proof:Let us consider that,tan A=x0ex0ex⇒cot(π/2 A)=x [∵tan A=cot(π/2 A)]0ex0ex⇒π/2 A=cot^ 1 x0ex0ex⇒cot^ 1 x+A=π/20ex0ex⇒cot^ 1 x+tan^ 1 x=π/2[∵ A=ta ...

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Standard X

Mathematics

Trigonometric Identities

tan −1 x+cot ...

Question

Prove that, $\tan^{- 1} x + \cot^{- 1} x = \frac{π}{2}, x \in R$

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Solution

Proof:
Let us consider that,
$\tan A = x \Rightarrow \cot (\frac{π}{2} - A) = x [∵ \tan A = \cot (\frac{π}{2} - A)] \Rightarrow \frac{π}{2} - A = \cot^{- 1} x \Rightarrow \cot^{- 1} x + A = \frac{π}{2} \Rightarrow \cot^{- 1} x + \tan^{- 1} x = \frac{π}{2} [∵ A = \tan^{- 1} x] ∴ \cot^{- 1} x + \tan^{- 1} x = \frac{π}{2}, x \in R$
Hence, proved.

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Prove that, tan-1x+cot-1x=π2,x∈R

Proof:Let us consider that,tanA=x⇒cotπ2-A=x∵tanA=cotπ2-A⇒π2-A=cot-1x⇒cot-1x+A=π2⇒cot-1x+tan-1x=π2∵A=tan-1x∴cot-1x+tan-1x=π2,x∈RHence, proved.

Prove that, $\tan^{- 1} x + \cot^{- 1} x = \frac{π}{2}, x \in R$