Question

I need help!! I will give 5.0 star ​

Answer 1

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