Question

Verify the following identities

Answer 1

See below for proof.

Explanation:

Use the following trigonometric identities to verify the given identities:

`\boxed{\cot(x)=(\cos(x))/(\sin(x))}`

`\boxed{\sec(x)=(1)/(\cos(x)) \implies \cos(x)=(1)/(\sec(x))}`

`\boxed{\tan(x)=(\sin(x))/(\cos(x))}`

`\boxed{\sin^2(x)+\cos^2(x)=1 \implies \cos^2(x)=1-\sin^2(x)}`

Question a)

`\begin{aligned}(1)/(\sin(x)\cot(x))&=(1)/(\sin(x) \cdot(\cos(x))/(\sin(x)))\\\\&=(1)/((\sin(x)\cos(x))/(\sin(x)))\\\\&=(\sin(x))/(\sin(x)\cos(x))\\\\&=(1)/(\cos(x))\end{aligned}`

Question b)

`\begin{aligned}\sec(x)-\tan(x)\sin(x)&=(1)/(\cos(x))-(\sin(x))/(\cos(x)) \cdot \sin(x)\\\\&=(1)/(\cos(x))-(\sin^2(x))/(\cos(x))\\\\&=(1-\sin^2(x))/(\cos(x))\\\\&=(\cos^2(x))/(\cos(x))\\\\&=\cos(x)\\\\&=(1)/(\sec(x))\end{aligned}`