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Solve the following : 1+sec203cot40

Answer
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Hint: Now to solve this question we need to simplify all the trigonometric functions using their formulas. We convert them into sine and cos functions using the formulas that secx=1cosx and cotx=cosxsinx . Now students will also need to use the formula that sin(ab)=sinacosbcosasinb to further simplify and solve this equation and find the necessary answer we need in this question.

Complete step-by-step answer:
Now here in this question we need to find the value of the expression 1+sec203cot40 . Now to find its value we can name the expression to be
E=1+sec203cot40
Now to solve this further we will start by making these equations in the form of sine and cos functions to make it simpler to solve. To convert the equation we use the formula of secant and cot that secx=1cosx and cotx=cosxsinx . By substituting these values in the above given expression we get
E=1+1cos203cos40sin40


Now we take the first and third term of this equation and taking its LCM we get
E=sin403cos40sin40+1cos20
Now in first term to simplify it further we multiply and divide both the numerator and denominator by 2 giving us
E=2(12sin4032cos40)sin40+1cos20
Now we know that we can write cos60=12 and we can also write sin60=32
E=2(cos60sin40sin60cos40)sin40+1cos20
Now we can see that the numerator of first term is in the form of sin(ab)=sinacosbcosasinb therefore we can see that a is 40 and b is 60. So we can write this equation is
E=2sin(4060)sin40+1cos20
We can also write sin40in the form of half angle of sine which gives us sin40=2sin20cos20 and we also know that negative angle of sine is always equal to negative of sine that is sin(x)=sinx; Therefore putting these values we get;
E=2sin202sin20cos20+1cos20
Now cancelling the common terms from both numerator and denominator
E=1cos20+1cos20
Therefore we get
E=0
So we can say that ; 1+sec203cot40=0

Note: To solve questions like this students should always try to remember the half angle and double angle formulas of functions. Converting them into sine and cos functions gives us a way to easily simplify these expressions. Students must also know the product and addition formulas of trigonometric functions. Another necessity required is to know the trigonometric values of all basic angles.