Question
Question: If \(\cos e{c^2}\theta + {\cot ^2}\theta = 7\), what is the value (in degrees) of \(\theta \)? (1)...
If cosec2θ+cot2θ=7, what is the value (in degrees) of θ?
(1) 15
(2) 30
(3) 45
(4) 60
Solution
We are given a trigonometric expression and we have to find the value of θ 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+tan2θ we will convert the tan2θ in the form of sinθ and cosθ and proceed it further accordingly and find the value of θ.In this case also we first convert the equation in sinθ and cosθ and solve it accordingly.
Complete step-by-step answer:
Step1: The given expression is cosec2θ+cot2θ=7 we will convert the expression in the simpler form of sinθ and cosθ.cosecθ=sinθ1 ;cotθ=sinθcosθ
So applying this we will get
Step2: Putting the values of cosecθ and cotθ we will get
⇒ sin2θ1+sin2θcos2θ=7
Taking L.C.M of sin2θ we get
⇒ sin2θcos2θ+1=7
Taking sin2θ in next side we will get
Step3: cos2θ+1=7sin2θ
Substitute the cos2θ=1−sin2θ we get:
⇒ 1−sin2θ+1=7sin2θ
Adding the like terms :
⇒ 2−sin2θ=7sin2θ
Rearranging the equation:
⇒ 82=sin2θ
Step4: Dividing 2 by 8 we get:
⇒ sin2θ=41
Taking square root both sides
⇒ sinθ=±21
⇒ sinθ=21 hence here θ=300 and
⇒ sinθ=−21 therefore θ=1500
Therefore θ=300 or θ=1500
Ignoring the value of θ=1500 as it is in negative quadrant we will take the value of θ=300
Final answer is θ=300.
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 sinθ and cosθ. 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: cos2θ+sin2θ=1
Given equations is: cosec2θ+cot2θ=7
Identity to remember: 1+cot2θ=cosec2θ
Substituting the value of cosec2θ we will get
⇒ 1+cot2θ+cot2θ=7
⇒ 1+2cot2θ=7
⇒ 2cot2θ=6
Dividing 6 by 2 we will get
cot2θ=3
On taking square root both the sides we get:
cotθ=±3
cotθ=3 hence here θ=300 and
cotθ=−3 therefore θ=1500
Hence value of θ will be 300 or θ=1500
Ignoring the value of θ=1500 as it is in negative quadrant we will take the value of θ=300