Question

If cos alpha + sec alpha = 2 show that cos ^ 8 alpha + sec ^ 8 alpha = 2

Answer 1

Answer:

First of all, in these type of questions, try to bring every trigonometric function or ratio in terms of sinθsin⁡θ and cosθcos⁡θ and sometimes even in tanθtan⁡θ.

Here we have cosθ+secθ=2cos⁡θ+sec⁡θ=2

So cosθ+1cosθ=2cos⁡θ+1cos⁡θ=2

Hence cos2θ+1=2cosθcos2⁡θ+1=2cos⁡θ

Or cos2θ−2cosθ+1=0cos2⁡θ−2cos⁡θ+1=0

Use the formula to find root of a quadratic equation. Here the equation is quadratic in cosθcos⁡θ

So cosθ=−(−2)±(−2)2−4(1)(1)−−−−−−−−−−−−√2(1)cos⁡θ=−(−2)±(−2)2−4(1)(1)2(1)

Hence cosθ=1cos⁡θ=1

Or θ=2nπ;θ=2nπ;for n=0,1,2,⋯n=0,1,2,⋯

Now we know that cosθcos⁡θ is always equal to 11 for 00 and even values of ππ and sinθsin⁡θ is always equal to 00 for even values of π