Trig simplification (1)

Geometry Level 3

Simplify the following fraction:

sec ⁡ α sec ⁡ β sec ⁡ γ 1 − tan ⁡ α tan ⁡ β − tan ⁡ α tan ⁡ γ − tan ⁡ β tan ⁡ γ \dfrac {\sec \alpha \sec \beta \sec \gamma}{1 - \tan \alpha \tan \beta - \tan \alpha \tan \gamma - \tan \beta \tan \gamma}

csc ⁡ ( α + β + γ ) \csc (\alpha + \beta + \gamma) None of the above cot ⁡ ( α + β + γ ) \cot (\alpha + \beta + \gamma) sec ⁡ ( α + β + γ ) \sec (\alpha + \beta + \gamma)

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Sharky Kesa by Sharky Kesa , 19, United Kingdom

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2 solutions

Multiply both the denominator and numerator with cosa.cosb.cosy. Then you can take cosy as common from the first two terms and (-siny) common from the last two terms of denominator. After some easy steps, you will get 1/cos (a+b+y) which equals sec (a+b+y)

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check for alpha=beta=gamma=45 its comes to be - 2 \sqrt{ 2} checking for the options we get sec(45+45+45)=sec(135)=- 2 \sqrt{ 2}

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