Question

If $\cos e{c^2}\theta + {\cot ^2}\theta = 7$, what is the value (in degrees) of $\theta $?
(1) $15$
(2) $30$
(3) $45$
(4) $60$

Answer 1

Hint: We are given a trigonometric expression and we have to find the value of $\theta $ here to solve this we will use the different trigonometric functions and identities. Here we will convert these expressions into simpler form. For example in case of identity $1 + {\tan ^2}\theta $ we will convert the ${\tan ^2}\theta $ in the form of $\sin \theta $ and $\cos \theta $ and proceed it further accordingly and find the value of $\theta $.In this case also we first convert the equation in $\sin \theta $ and $\cos \theta $ and solve it accordingly.

Complete step-by-step answer:
Step1: The given expression is $\cos e{c^2}\theta + {\cot ^2}\theta = 7$ we will convert the expression in the simpler form of $\sin \theta $ and $\cos \theta $.$\cos ec\theta = \dfrac{1}{{\sin \theta }}$ ;$\cot \theta = \dfrac{{\cos \theta }}{{\sin \theta }}$
So applying this we will get
Step2: Putting the values of $\cos ec\theta $ and $\cot \theta $ we will get
$ \Rightarrow $ $\dfrac{1}{{{{\sin }^2}\theta }} + \dfrac{{{{\cos }^2}\theta }}{{{{\sin }^2}\theta }} = 7$
Taking L.C.M of ${\sin ^2}\theta $ we get
$ \Rightarrow $$\dfrac{{{{\cos }^2}\theta + 1}}{{{{\sin }^2}\theta }} = 7$
Taking ${\sin ^2}\theta $ in next side we will get
Step3: ${\cos ^2}\theta + 1 = 7{\sin ^2}\theta $
Substitute the ${\cos ^2}\theta = 1 - {\sin ^2}\theta $ we get:
$ \Rightarrow $$1 - {\sin ^2}\theta + 1 = 7{\sin ^2}\theta $
Adding the like terms :
$ \Rightarrow $$2 - {\sin ^2}\theta = 7{\sin ^2}\theta $
Rearranging the equation:
$ \Rightarrow $$\dfrac{2}{8} = {\sin ^2}\theta $
Step4: Dividing $2$ by $8$ we get:
$ \Rightarrow $${\sin ^2}\theta = \dfrac{1}{4}$
Taking square root both sides
$ \Rightarrow $$\sin \theta = \pm \dfrac{1}{2}$
$ \Rightarrow $$\sin \theta = \dfrac{1}{2}$ hence here $\theta = {30^0}$ and
$ \Rightarrow $$\sin \theta = - \dfrac{1}{2}$ therefore $\theta = {150^0}$
Therefore $\theta $=${30^0}$ or $\theta = {150^0}$
Ignoring the value of $\theta = {150^0}$ as it is in negative quadrant we will take the value of $\theta = {30^0}$
Final answer is $\theta $=${30^0}$.

Option B is the correct answer.

Note: In this students mainly get confused in applying the identities or converting the expression into simpler forms of $\sin \theta $ and $\cos \theta $. They also get confused in solving equations so formed. In such questions students first solve the part which requires calculations and solve it according to the need of the equation. We can also solve this by using other method identity used: ${\cos ^2}\theta + {\sin ^2}\theta = 1$
Given equations is: $\cos e{c^2}\theta + {\cot ^2}\theta = 7$
Identity to remember: $1 + {\cot ^2}\theta = \cos e{c^2}\theta $
Substituting the value of $\cos e{c^2}\theta $ we will get
$ \Rightarrow $$1 + {\cot ^2}\theta + {\cot ^2}\theta = 7$
$ \Rightarrow $$1 + 2{\cot ^2}\theta = 7$
$ \Rightarrow $$2{\cot ^2}\theta = 6$
Dividing $6$ by $2$ we will get
${\cot ^2}\theta = 3$
On taking square root both the sides we get:
$\cot \theta = \pm \sqrt 3 $
$\cot \theta = \sqrt 3 $ hence here $\theta = {30^0}$ and
 $\cot \theta = - \sqrt 3 $ therefore $\theta = {150^0}$
Hence value of $\theta $ will be ${30^0}$ or $\theta = {150^0}$
Ignoring the value of $\theta = {150^0}$ as it is in negative quadrant we will take the value of $\theta = {30^0}$