Question

differentiate cos-1 (2coax + 3sinx/√13)
.
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please tell me the correct answer
because it's my first unit test paper for class 12th
don't tell me wrong answer​

Answer 1

Answer:

Step-by-step explanation:

We have,

$\tt{y=cos^{-1}\left(\dfrac{2\,cos(x)+3\,sin(x)}{\sqrt{13}}\right)}$

$\tt{\implies\,y=cos^{-1}\left\{\dfrac{2}{\sqrt{13}}\,cos(x)+\dfrac{3}{\sqrt{13}}\,sin(x)\right\}}$

$\sf{Let\,\,\,\blue{cos(\alpha)=\dfrac{2}{\sqrt{13}}}}$

$\sf{\blue{\implies\,sin(\alpha)=\sqrt{1-cos^{2}(\alpha)}}}$

$\sf{\blue{\implies\,sin(\alpha)=\sqrt{1-\left(\dfrac{2}{\sqrt{13}}\right)^{2}}}}$

$\sf{\blue{\implies\,sin(\alpha)=\sqrt{1-\dfrac{4}{13}}}}$

$\sf{\blue{\implies\,sin(\alpha)=\sqrt{\dfrac{13-4}{13}}}}$

$\sf{\blue{\implies\,sin(\alpha)=\sqrt{\dfrac{9}{13}}}}$

$\sf{\blue{\implies\,sin(\alpha)=\dfrac{3}{\sqrt{13}}}}$

Now,

$\tt{\implies\,y=cos^{-1}\left\{cos(\alpha)\,cos(x)+sin(\alpha)\,sin(x)\right\}}$

$\tt{\implies\,y=cos^{-1}\left\{cos(\alpha-x)\right\}}$

$\tt{\implies\,y=\alpha-x}$

$\tt{\implies\,\dfrac{dy}{dx}=-1}$