Question

Prove it. this question is from trigonometry ​

Answer 1

See below for step-by-step proof.

Explanation:

`\textsf{Given expression}:`

`\cos^6 \theta+\sin^6 \theta`

`\textsf{Apply exponent rule} \quad a^(bc)=(a^b)^c:`

`\implies (\cos^2 \theta)^3+(\sin^2 \theta)^3`

`\textsf{Use the identity}\quad a^3+b^3=(a+b)^3-3ab(a+b):`

`\implies (\cos^2 \theta+\sin^2 \theta)^3-3 \cos^2 \theta\sin^2 \theta (\cos^2 \theta+\sin^2 \theta)`

`\textsf{Use the identity}\quad \sin^2 \theta + \cos^2 \theta=1:`

`\implies (1)^3-3 \cos^2 \theta\sin^2 \theta (1)`

`\implies 1-3 \cos^2 \theta\sin^2 \theta`

`\implies 1-3 (\cos \theta\sin \theta)^2`

`\textsf{Use the identity}\quad \sin 2 \theta=2 \sin \theta \cos \theta \implies (1)/(2)\sin 2 \theta= \sin \theta \cos \theta:`

`\implies 1-3 \left( (1)/(2)\sin 2 \theta\right)^2`

`\implies 1-(3)/(4) \sin^2 2 \theta`

`\textsf{Use the identity}\quad \sin^2 2\theta + \cos^2 2\theta=1 \implies \sin^2 2\theta=1-\cos^2 2\theta`

`\implies 1-(3)/(4) \left(1-\cos^2 2 \theta\right)`

`\implies 1-(3)/(4) +(3)/(4)\cos^2 2 \theta`

`\implies (1)/(4) +(3)/(4)\cos^2 2 \theta`

`\textsf{Factor out }(1)/(4):`

`\implies (1)/(4)\left(1+3\cos^2 2 \theta\right)`

`\textsf{Use the identity}\quad \cos (4 \theta)=2 \cos^2 2\theta - 1 \implies \cos^2 2 \theta=(1)/(2)\cos 4 \theta+(1)/(2):`

`\implies (1)/(4)\left(1+3\left((1)/(2)\cos 4 \theta+(1)/(2)\right)\right)`

`\implies (1)/(4)\left(1+(3)/(2)\cos 4 \theta+(3)/(2)\right)`

`\implies (1)/(4)\left((5)/(2)+(3)/(2)\cos 4 \theta\right)`

`\implies (1)/(4) \cdot (1)/(2)\left(5+3\cos 4 \theta\right)`

`\implies (1)/(8)\left(5+3\cos 4 \theta\right)`