Maths - Please Help Me?

Question

Hello,

There are some questions that I am not sure how to do:

1) (sinA) / (sinB) - (cosA) / (cosB) = [2sin(A - B)] / [sin2B]

2) tanA + cotA = 2cosec2A

3) cotA - tanA = 2cot2A

Thank you very much.

Answer 1

1) (sinA) / (sinB) - (cosA) / (cosB) = [2sin(A - B)] / [sin2B]

Left hand side identity : (sinA) / (sinB) - (cosA) / (cosB)

Take LCM sinBcosB.

(sinA) / (sinB) - (cosA) / (cosB)=(sinAcosB-cosAsinB)/sinBcosB

Composite formula :sin(A - B) = sin A cos B - cos A sin B.

=sin(A - B) /sinBcosB

Multiply numerator and denominator by 2.

= 2sin(A - B) / 2sin B cos B

Double angle formula : sin(2B) = 2sin B cos B.

= 2sin(A - B) / sin (2B)

= Right hand side identity.

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2) tanA + cotA = 2cosec2A

Left hand side identity : tanA + cotA

Use formulas: tanA =sinA/cosA;cotA = cosA/sinA

tanA + cotA =sinA/cosA+cosA/sinA

                  =(sinAsinA+cosAcosA)/cosAsinA

                   =(sin²A+cos²A)/cosAsinA

Pythagorean identity : sin² A + cos² A = 1.

=1/cosAsinA

Multiply numerator and denominator by 2.

=2/2cosAsinA

Double angle formula : sin(2A) = 2sin A cos A.

=2/sin2A

=2cosec2A

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3) cotA - tanA = 2cot2A

Left hand side identity : cotA - tanA

Use formulas: tanA =sinA/cosA;cotA = cosA/sinA

cotA - tanA =cosA/sinA-sinA/cosA

                  =(cosAcosA-sinAsinA)/cosAsinA

                   =(cos²A-sin²A)/cosAsinA

Double angle formula : cos(2A) = cos² A - sin² A.

=cos2A/cosAsinA

Multiply numerator and denominator by 2.

=2cos2A/2cosAsinA

Double angle formula : sin(2A) = 2sin A cos A.

=2cos2A/sin2A

=2cot2A