Playing with Trigs

Geometry Level 2

If $\sin\theta + \csc\theta = 2$ then find the value of

$\sin^n\theta + \csc^n\theta$

The answer is 2.

1 solution

Viki Zeta
Sep 10, 2016

$sin\theta + cosec\theta = 2 \\ sin\theta + \dfrac{1}{sin\theta} = 2 \\ \dfrac{sin^2\theta + 1}{sin\theta} = 2 \\ sin^2\theta + 1 = 2sin\theta \\ sin^2\theta - 2sin\theta + 1 = 0\\ (1 - sin\theta)(1-sin\theta) = 0\\ sin\theta = 1 \\ sin^n\theta + cosec^n\theta = (sin\theta)^n + \dfrac{1}{(sin\theta)^n} = 1 + \dfrac{1}{1} = 2$

Though I don't encourage such methods, but this is what one should do in objective:

Plug $n=1$ and see the magic ;)

Swapnil Das - 4 years, 9 months ago

No this won't always work. This will work only with given condition. Take for example $\theta=0$ . Then the answer would be undefined

Viki Zeta - 4 years, 8 months ago

That's why I don't encourage such methods :P

Swapnil Das - 4 years, 8 months ago

@Swapnil Das – Did you read my reply? Unless you use the condition (as I solved), your answer would be wrong. Because taking any value need not be 100% satisfying given condition.

Viki Zeta - 4 years, 8 months ago

@Viki Zeta – ok, fine. Be sure to read my reply carefully, also.

Swapnil Das - 4 years, 8 months ago

Playing with Trigs

The answer is 2.

1 solution

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