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Easy

Question

If sin space left parenthesis alpha plus beta right parenthesis equals 1 comma sin space left parenthesis alpha minus beta right parenthesis equals 1 half then tan space left parenthesis alpha plus 2 beta right parenthesis tan space left parenthesis 2 alpha plus beta right parenthesis equals

  1. 1
  2. -1
  3. 0
  4. 1 half

hintHint:

When we find sin cos and tan values for a triangle, we usually consider these angles: 0°, 30°, 45°, 60° and 90°. It is easy to memorise the values for these certain angles. The trigonometric values are about the knowledge of standard angles for a given triangle as per the trigonometric ratios (sine, cosine, tangent, cotangent, secant and cosecant).

The correct answer is: 1


    If sin space left parenthesis alpha plus beta right parenthesis equals 1 comma sin space left parenthesis alpha minus beta right parenthesis equals 1 half then tan space left parenthesis alpha plus 2 beta right parenthesis tan space left parenthesis 2 alpha plus beta right parenthesis
    rightwards double arrow s in space left parenthesis alpha plus beta right parenthesis equals 1
W e space k n o w space t h a t space semicolon space sin 90 degree space equals space 1 space s o comma space space left parenthesis alpha plus beta right parenthesis space equals space 90 degree
rightwards double arrow sin space left parenthesis alpha minus beta right parenthesis equals 1 half
W e space k n o w space t h a t space semicolon space sin 30 degree space equals space 1 half space s o comma space space left parenthesis alpha minus beta right parenthesis space equals space 30 degree
F r o m space t h e space a b o v e space w e space g e t comma
space alpha space equals space 60 degree comma space beta space equals space 30 degree
N o w comma bold space bold italic t bold italic a bold italic n bold left parenthesis bold 2 bold italic alpha bold plus bold italic beta bold right parenthesis bold space bold italic t bold italic a bold italic n bold left parenthesis bold italic alpha bold plus bold 2 bold italic beta bold right parenthesis
t a n left parenthesis 2 alpha plus beta right parenthesis space t a n left parenthesis alpha plus 2 beta right parenthesis space equals space t a n left parenthesis 150 degree right parenthesis space t a n left parenthesis 120 degree right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space space c o t left parenthesis 60 degree right parenthesis space c o t left parenthesis 30 degree right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 1 over denominator square root of 3 end fraction cross times square root of 3 space equals space 1

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