Question

If $\sin \theta + \cos ec\theta = 2$, then the value of ${\sin ^{10}}\theta + \cos e{c^{10}}\theta $ is
(a) $2$
(b) $210$
(c) $29$
(d) $10$

Answer 1

Hint: We start solving the problem by using trigonometric property $\cos ec\theta = \dfrac{1}{{\sin \theta }}$. We then start solving the equation formed make the necessary calculations to get the required value of ${\sin ^{10}}\theta + \cos e{c^{10}}\theta $.

Formula Used: $\cos ec\theta = \dfrac{1}{{\sin \theta }}$

Complete step by step Solution:
According to the problem, we are given that the sum of the trigonometric functions as $\sin \theta + \cos ec\theta = 2$ and we need to find the value of ${\sin ^{10}}\theta + \cos e{c^{10}}\theta $
Now, we have $\sin \theta + \cos ec\theta = 2$---(1)
Let substitute $\cos ec\theta = \dfrac{1}{{\sin \theta }}$ in equation (1)
$ \Rightarrow \sin \theta + (\dfrac{1}{{\sin \theta }}) - 2 = 0$
$ \Rightarrow {\sin ^2}\theta + 1 - 2sin\theta = 0$
$ \Rightarrow {(\sin \theta - 1)^2} = 0$
$ \Rightarrow \sin \theta = 1$
Let us now find the value of $\cos ec\theta $
So, we have $\cos ec\theta = \dfrac{1}{{\sin \theta }}$

$ \Rightarrow \cos ec\theta = \dfrac{1}{{\sin \theta }} = 1$
Now, let us find the value of ${\sin ^{10}}\theta + \cos e{c^{10}}\theta $
So, we have ${\sin ^{10}}\theta + \cos e{c^{10}}\theta = {1^{10}} + {1^{10}}$
$ \Rightarrow {\sin ^{10}}\theta + \cos e{c^{10}}\theta = 2$
∴ We have found the value of ${\sin ^{10}}\theta + \cos e{c^{10}}\theta = 2$

Hence, the correct option is (a).

Additional Information
Trigonometry is the branch of Mathematics in which we deal with specific functions of angles, their applications, and their calculations. In Mathematics, there are a total of six different types of Trigonometric functions:
Sine ($\sin $)
Cosine ($\cos $)
Secant ($\sec $)
Cosecant ($\cos ec$)
Tangent ($\tan $)
Cotangent ($\cot $)
They symbolize the relationship between the ratios of different sides of a right-angle Triangle. These Trigonometric functions can also be called circular functions as their values can be written as the ratios of x and y coordinates of the circle of Radius 1 that keep in touch with angles in standard positions.

Note: We can also solve the problem by squares on both sides of eq. 1 but it would be a lengthy calculation. We can also solve this problem by performing a trial and error method for the angle in $\sin \theta + \cos ec\theta = 2$. Similarly, we can expect problems to find the value of ${\sin ^m}\theta + \cos e{c^m}\theta $ using the obtained value of $\sin \theta $ and $\cos ec\theta $.