Question

Help please!

I need to prove this using identities
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cos(pi/2+x)/cos(pi+x)=tanx​

Answer 1

Answer: see proof below

Explanation:

Use the Sum & Difference Identity: cos (A + B) = cos A · cos B - sin A · sin B

Recall the following from Unit Circle: cos (π/2) = 0, sin (π/2) = 1

cos (π) = -1, sin (π) = 0

Use the Quotient Identity:
`\tan A=(\sin A)/(\cos A)`

Proof LHS → RHS:

`\text{LHS:}\qquad \qquad (\cos \bigg((\pi)/(2)+x\bigg))/(\cos \bigg(\pi +x\bigg))`

`\text{Sum Difference:}\qquad (\cos (\pi)/(2)\cdot \cos x-\sin (\pi)/(2)\cdot \sin x)/(\cos \pi \cdot \cos x-\sin \pi \cdot \sin x)`

`\text{Unit Circle:}\qquad \qquad (0\cdot \cos x-1\cdot \sin x)/(-1\cdot \cos x-0\cdot \sin x)`

`=(-\sin x)/(-\cos x)`

Quotient: tan x

LHS = RHS
`\checkmark`