Question

Use trigonometric identities to verify each expression is equal.

c) csc^2(x)-2csc(x)cot(x)+cot^2(x) = tan^2(x/2)

d) [cos(x)cos(y)][tan(x)+tan(y)] = sin(x+y)

Answer 1

Part (c)

`\text{LHS}=\csc^(2) x-2\csc x\cot x+\cot^(2) x\\\\=(\csc x-\cot x)^(2)\\\\=\left((1)/(\sin x)-(\cos x)/(\sin x) \right)^(2)\\\\=\left((1-\cos x)/(\sin x))^(2)\\\\=\tan^(2) \left((x)/(2) \right)\\\\=\text{RHS}`

Part (d)

`\text{LHS}=[\cos x \cos y][\tan x+\tan y]\\\\=[\cos x\cos y]\left[(\sin x)/(\cos x)+(\sin y)/(\cos y) \right]\\\\=\sin x \cos y+\sin y \cos x\\\\=\sin(x+y)\\\\=\text{RHS}`