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Question: How do you simplify: \( \left( {1 + {{\tan }^2}x} \right)\left( {{{\cos }^2}x} \right) \) ....

How do you simplify: (1+tan⁡2x)(cos⁡2x)\left( {1 + {{\tan }^2}x} \right)\left( {{{\cos }^2}x} \right) .

Explanation

Solution

Hint : The given question deals with basic simplification of trigonometric functions by using some of the simple trigonometric formulae such as tan⁡(x)=sin⁡xcos⁡x\tan (x) = \dfrac{{\sin x}}{{\cos x}} and cot⁡(x)=cos⁡xsin⁡x\cot (x) = \dfrac{{\cos x}}{{\sin x}} . Basic algebraic rules and trigonometric identities are to be kept in mind while doing simplification in the given problem.

Complete step-by-step answer :
In the given problem, we have to simplify the product of cos⁡2x{\cos ^2}x and (1+tan⁡2x)\left( {1 + {{\tan }^2}x} \right) .
So, (1+tan⁡2x)(cos⁡2x)\left( {1 + {{\tan }^2}x} \right)\left( {{{\cos }^2}x} \right)
Now, we express the trigonometric expression in sine and cosine using tan⁡(x)=sin⁡xcos⁡x\tan (x) = \dfrac{{\sin x}}{{\cos x}} ,
⇒\Rightarrow (1+sin⁡2xcos⁡2x)(cos⁡2x)\left( {1 + \dfrac{{{{\sin }^2}x}}{{{{\cos }^2}x}}} \right)\left( {{{\cos }^2}x} \right)
Taking LCM of terms inside the bracket, we get,
⇒\Rightarrow (cos⁡2x+sin⁡2xcos⁡2x)(cos⁡2x)\left( {\dfrac{{{{\cos }^2}x + {{\sin }^2}x}}{{{{\cos }^2}x}}} \right)\left( {{{\cos }^2}x} \right)
Now, we know that cos⁡2x+sin⁡2x{\cos ^2}x + {\sin ^2}x is equal to one from trigonometric identity. So, simplifying the expression, we get,
⇒\Rightarrow (1cos⁡2x)(cos⁡2x)\left( {\dfrac{1}{{{{\cos }^2}x}}} \right)\left( {{{\cos }^2}x} \right)
Cancelling the common factors in numerator and denominator, we get,
⇒\Rightarrow 11
Hence, the product (1+tan⁡2x)(cos⁡2x)\left( {1 + {{\tan }^2}x} \right)\left( {{{\cos }^2}x} \right) can be simplified as 11 by the use of basic algebraic rules and simple trigonometric formulae.
So, the correct answer is “1”.

Note : Given problem deals with Trigonometric functions. For solving such problems, trigonometric formulae should be remembered by heart such as: tan⁡(x)=sin⁡(x)cos⁡(x)\tan (x) = \dfrac{{\sin (x)}}{{\cos (x)}} and cot⁡(x)=cos⁡(x)sin⁡(x)\cot (x) = \dfrac{{\cos (x)}}{{\sin (x)}} . Besides these simple trigonometric formulae, trigonometric identities are also of significant use in such type of questions where we have to simplify trigonometric expressions with help of basic knowledge of algebraic rules and operations. However, questions involving this type of simplification of trigonometric ratios may also have multiple interconvertible answers.