Prove that \[\frac{{\cos \theta {\text{ }}}}{{\sec \theta - \tan \theta }} = {\text{ }}1 + \sin \theta \]

Answer

A

Explanation

Let `f(x)=sin^(2)x and g(x)=sqrt(x)`
Now, `fog(x)=f[g(x)] = f(sqrt(x))=sin^(2) sqrt(x)`
` and gof(x)=g[f(x)]=g(sin^(2)x)=sqrt(sin^(2)x)=|sinx|`
Again, let `f(x)=sinx, g(x) =|x|`
`fog(x)=f[g(x)]=f(|x|)`
`=sin|x| ne (sin sqrt (x))^(2)`
When ` f(x) = x^(2), g(x)=sin sqrt(x)`
`fog(x) = f[g(x)]=f(sin sqrt(x))=(sin sqrt(x))^(2)`
`and (gof) (x) = g[f(x)]=g(x^(2))=sin sqrt(x^(2))`
`=sin|x| ne |sinx|`