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Question: Simplify \(\dfrac{{{{\sin }^2}\theta + {{\cos }^2}\theta }}{{{{\cos }^2}\theta }}\)....

Simplify sin⁡2θ+cos⁡2θcos⁡2θ\dfrac{{{{\sin }^2}\theta + {{\cos }^2}\theta }}{{{{\cos }^2}\theta }}.

Explanation

Solution

The given problem requires us to simplify the given trigonometric expression. The question requires thorough knowledge of trigonometric functions, formulae and identities. The question describes the wide ranging applications of trigonometric identities and formulae. We must keep in mind the trigonometric identities while solving such questions.

Complete step by step solution:
In the given question, we are required to evaluate the value of sin⁡2θ+cos⁡2θcos⁡2θ\dfrac{{{{\sin }^2}\theta + {{\cos }^2}\theta }}{{{{\cos }^2}\theta }} using the basic concepts of trigonometry and identities.

So, using the trigonometric identity sin⁡2θ+cos⁡2θ=1{\sin ^2}\theta + {\cos ^2}\theta = 1, we get

== 1cos⁡2θ\dfrac{1}{{{{\cos }^2}\theta }}

Now, we know that sec⁡(θ)\sec \left( \theta \right) is reciprocal of cos⁡(θ)\cos \left( \theta \right). So, replacing 1cos⁡θ\dfrac{1}{{\cos \theta }} by sec⁡(θ)\sec \left( \theta \right) .

== sec⁡2θ{\sec ^2}\theta

So, we get the value of trigonometric expression sin⁡2θ+cos⁡2θcos⁡2θ\dfrac{{{{\sin }^2}\theta + {{\cos }^2}\theta }}{{{{\cos }^2}\theta }} as sec⁡2θ{\sec ^2}\theta .

These identities are of vital importance for solving any question involving trigonometric functions and identities. All the trigonometric ratios can be converted into each other using the simple trigonometric identities listed above.

Note: The given problem involves the use of trigonometric formulae and identities. Such questions require thorough knowledge of trigonometric conversions and ratios. Algebraic operations and rules like transposition rule come into significant use while solving such problems.