
Show that the values of .
Answer
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Hint: In this question we have to show that the given trigonometric expression equals zero. For this we are going to show that by using trigonometric identities in angle and ratio and also we are going to multiply and add the trigonometric identities in complete step by step solution.
Trigonometric is a function that deals with the relationship between the sides and angles of triangles.
Formula used: There are six function of an angle commonly used in trigonometry, they are . In this sum we are going to see about only and angle and ratio formula. The formulas are
Complete step-by-step answer:
Let consider the given equation as .
Now, Rewrite the above equation as .
Here, we applying the trigonometric angles formulas on the expression, the and angle are commonly known as and .
Now, we get
and
Substitute the two trigonometric identities into the , we get
Taking as a common term in the above both terms, then
Now, cross multiply the into denominator of right hand side, we get
We know that any number divisible by zero is zero. Then,
Applying the trigonometry identity,
Now, consider here, and . Then, we substitute in the above trigonometric identity, we get
After adding and subtracting the trigonometric identities, we get
Since there are two solutions:
1. if and only if and .
2.
Now, we have to find the angle where the sine term becomes zero.
.
Also find another angle for sine term becomes zero.
.
if and only if and .
Therefore, from the above angle which shows that the trigonometric identities become zero.
Hence proved.
Note: All the trigonometric functions are positive in the first quadrant. SIn and Cosec are positive in the second quadrant. Tan and Cot are positive in the third quadrant. Cos and Sec are positive in the fourth quadrant.
Trigonometric is a function that deals with the relationship between the sides and angles of triangles.
Formula used: There are six function of an angle commonly used in trigonometry, they are
Complete step-by-step answer:
Let consider the given equation as
Now, Rewrite the above equation as
Here, we applying the trigonometric angles formulas on the expression, the
Now, we get
Substitute the two trigonometric identities into the
Taking
Now, cross multiply the
We know that any number divisible by zero is zero. Then,
Applying the trigonometry identity,
Now, consider here,
After adding and subtracting the trigonometric identities, we get
Since there are two solutions:
1.
2.
Now, we have to find the angle where the sine term becomes zero.
Also find another angle for sine term becomes zero.
Therefore, from the above angle which shows that the trigonometric identities become zero.
Hence proved.
Note: All the trigonometric functions are positive in the first quadrant. SIn and Cosec are positive in the second quadrant. Tan and Cot are positive in the third quadrant. Cos and Sec are positive in the fourth quadrant.

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