Question

Find the value of sin²theta +1/1+tan²theta.
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Answer 1

Given : $\sf{\bf{ \sin^{2} \theta +\dfrac{1}{1+\tan^{2}\theta}}}$

To Find : The value of $\sf{ \sin^{2} \theta +\dfrac{1}{1+\tan^{2}\theta}}$

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⠀⠀⠀⠀⠀ $\sf{\longrightarrow{ \sin^{2} \theta +\dfrac{1}{1+\tan^{2}\theta}}}$

As , We know that ,

$\star 1 + tan^{2}\theta = sec^{2} \theta$

⠀⠀⠀⠀⠀ $\sf{\longrightarrow{ \sin^{2} \theta +\dfrac{1}{\purple {1+\tan^{2}\theta}}}}$

⠀⠀⠀⠀⠀ $\sf{\longrightarrow{ \sin^{2} \theta +\dfrac{1}{\purple {\sec^{2}\theta}}}}$

⠀⠀⠀⠀⠀ $\sf{\longrightarrow{ \sin^{2} \theta +\dfrac{1} {\sec^{2}\theta}}}$

As , We know that ,

$\star \dfrac{1}{\sec^{2}\theta}= \cos^{2} \theta$

⠀⠀⠀⠀⠀ $\sf{\longrightarrow{ \sin^{2} \theta +\purple {\dfrac{1}{\sec^{2}\theta}}}}$

⠀⠀⠀⠀⠀ $\sf{\longrightarrow{ \sin^{2} \theta +\purple {\cos^{2}\theta}}}$

⠀⠀⠀⠀⠀ $\sf{\longrightarrow{ \sin^{2} \theta +\cos^{2}\theta}}$

As , We know that ,

$\star \sin^{2}\theta+ \cos^{2} \theta= 1$

⠀⠀⠀⠀⠀ $\sf{\longrightarrow{ \purple {\sin^{2} \theta +\cos^{2}\theta}}}$

⠀⠀⠀⠀⠀ $\implies \underline {\boxed {\sf{\purple {\quad 1\quad}}}}$

⠀⠀ $\underline {\dag {\sf{\pink {Hence,\:The \:value \:of\: \sin^{2} \theta +\dfrac{1}{1+\tan^{2}\theta}\:is\:1. }}}}$

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$\large{\boxed {\sf |\:\:\:{\underline {More \:To\;Know\::}}\:\:\:|}}$

Trigonometric Identities :

$\boxed{\begin{minipage}{6cm} Important Trigonometric identities :- \\ \\ \: \: 1)\:\sin^2\theta+\cos^2\theta=1 \\ \\ 2)\:\sin^2\theta= 1-\cos^2\theta \\ \\ 3)\:\cos^2\theta=1-\sin^2\theta \\ \\ 4)\:1+\cot^2\theta=\text{cosec}^2 \, \theta \\ \\5)\: \text{cosec}^2 \, \theta-\cot^2\theta =1 \\ \\ 6)\:\text{cosec}^2 \, \theta= 1+\cot^2\theta \\\ \\ 7)\:\sec^2\theta=1+\tan^2\theta \\ \\ 8)\:\sec^2\theta-\tan^2\theta=1 \\ \\ 9)\:\tan^2\theta=\sec^2\theta-1 \end{minipage}}$

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