Chapter 7.1, Problem 42E

To determine

To verify the identity:

  ${\left(\sin x+\cos x\right)}^4={\left(1+2\sin x\cos x\right)}^2$

Explanation of Solution

Given:

An identity is given as

  ${\left(\sin x+\cos x\right)}^4={\left(1+2\sin x\cos x\right)}^2$

Concept Used:

The trigonometric identities:

  ${\sin }^{2}x+{\cos }^{2}x=1$

Algebraic identity

  ${\left(a+b\right)}^2=a^2+2ab+b^2$

Calculation:

In order to verify the identity:

  ${\left(\sin x+\cos x\right)}^4={\left(1+2\sin x\cos x\right)}^2$

Take left hand of above identity and simplify it using the trigonometric identities.

As,

  $\begin{array}{c}{\left(\sin x+\cos x\right)}^4={\left[{\left(\sin x+\cos x\right)}^2\right]}^2\\ ={\left[{\left(\sin x\right)}^2+{\left(\cos x\right)}^2+2\left(\sin x\right)\left(\cos x\right)\right]}^2\\ ={\left({\sin }^{2}x+{\cos }^{2}x+2\sin x\cos x\right)}^2\\ ={\left(1+2\sin x\cos x\right)}^2\\ =\text{Right hand}\end{array}$

Thus, identity is verified, i.e.

  ${\left(\sin x+\cos x\right)}^4={\left(1+2\sin x\cos x\right)}^2$