Question

Prove that sin2a/1+cos2a=tana and deduce the values of tan 15 and tan22½

Answer 1

Step-by-step explanation:

Given Prove that sin2a/1+cos2a=tana and deduce the values of tan 15 and tan 22½

• Given  sin 2a / 1 + cos 2a------------1
• We know that sin 2a = 2 sin a cos a
• We know that cos 2a = 2cos^2 a – 1
• Substituting in 1 we get
•         2 sin a cos a / 1 + 2 cos^2 a - 1
•            2sin a cos a / 2 cos^2 a
•             2 sin a cos a / 2 cos a cos a
•                  So we get tan a
• Now tan 15 degree = tan (45 – 30)
• Now tan (a – b) = tan a – tan b / 1 – tan a tan b
• So tan (45 – 30) = tan 45 – tan 30 / 1 – tan 45 tan 30
•                           = 1 – 1/ √3 / 1+ 1 / √3
• Therefore tan 15 = √3 – 1 / √3 + 1
• Also tan 22 1/2 degree = √1 – cos 45 degree / 1 + cos 45 degree
•                                        = √1 -  1/√2 / 1 + 1/√2
•                                        = √√2 – 1 / √2 + 1
• Rationalising the denominator we get
•                                            = √2 – 1 / √2 + 1 ,√2 – 1 / √2 – 1
•                                           = √(√2 – 1)^2 / 2 – 1)
•               So tan 22 ½ degree = √2 - 1

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