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I need help!! I will give 5.0 star ​

I need help!! I will give 5.0 star ​-example-1
User Chlunde
by
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1 Answer

10 votes

So, here we need to differentiate tan³
(\theta), wr.t.
\theta, but let's recall some identities which will be very useful in this question :


  • {\boxed{\bf{(d)/(dx)\{\tan (x)\}=\sec^(2)(x)}}}


  • {\boxed{\bf{(d)/(dx)(x^n)=nx^(n-1)}}}


  • {\boxed{\bf{\sec^(2)(\theta)=\tan^(2)(\theta)+1}}}

Coming back on the question, consider :


{:\implies \quad \sf (d)/(d\theta)\{\tan^(3)(\theta)\}}


{:\implies \quad \sf 3\tan^(3-1)(\theta)(d)/(d\theta)\{\tan (\theta)\}}


{:\implies \quad \sf 3\tan^(2)(\theta)\sec^(2)(\theta)}

Using the identity ;


{:\implies \quad \sf 3\tan^(2)(\theta)\{1+\tan^(2)(\theta)\}}


{:\implies \quad \sf 3\tan^(2)(\theta)+3tan^(4)(\theta)}


{:\implies \quad \sf 3\tan^(4)(\theta)+3\{\sec^(2)(\theta)-1\}}


{:\implies \quad \boxed{\bf 3\tan^(4)(\theta)+3\sec^(2)(\theta)-3}}

Hence, Proved

User Gianni Spear
by
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