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Question: How do you simplify \(\dfrac{{{{\cos }^2}\left( {\dfrac{\pi }{2} - x} \right)}}{{\cos x}}\) ?...

How do you simplify cos⁡2(π2−x)cos⁡x\dfrac{{{{\cos }^2}\left( {\dfrac{\pi }{2} - x} \right)}}{{\cos x}} ?

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 answer:
In the given question, we are required to evaluate the value of cos⁡2(π2−x)cos⁡x\dfrac{{{{\cos }^2}\left( {\dfrac{\pi }{2} - x} \right)}}{{\cos x}} using the basic concepts of trigonometry and identities.So, using the trigonometric identity cos⁡(π2−x)=sin⁡x\cos \left( {\dfrac{\pi }{2} - x} \right) = \sin x, we get,
sin⁡2xcos⁡x\dfrac{{{{\sin }^2}x}}{{\cos x}}
Now, we know that tan⁡(θ)\tan \left( \theta \right) is ratio of sin⁡(θ)\sin \left( \theta \right) and cos⁡(θ)\cos \left( \theta \right). So, replacing sin⁡θcos⁡θ\dfrac{{\sin \theta }}{{\cos \theta }} by tan⁡(θ)\tan \left( \theta \right), we get,
tan⁡(x)sin⁡(x)\tan \left( x \right)\sin \left( x \right)

Hence, we get the value of trigonometric expression cos⁡2(π2−x)cos⁡x\dfrac{{{{\cos }^2}\left( {\dfrac{\pi }{2} - x} \right)}}{{\cos x}} as tan⁡(x)sin⁡(x)\tan \left( x \right)\sin \left( x \right).

Additional Information:
There are six trigonometric ratios: sin⁡θ\sin \theta , cos⁡θ\cos \theta , tan⁡θ\tan \theta , cos⁡ecθ\cos ec\theta , sec⁡θ\sec \theta and cot⁡θ\cot \theta . cos⁡ecθ\cos ec\theta is reciprocal of sin⁡θ\sin \theta . Similarly, sec⁡θ\sec \theta is reciprocal of cos⁡θ\cos \theta . tan⁡θ\tan \theta is ratio of sin⁡θ\sin \theta to cos⁡θ\cos \theta . Also,cot⁡θ\cot \theta is the reciprocal of tan⁡θ\tan \theta . Hence, cot⁡θ\cot \theta is the ratio of cos⁡θ\cos \theta to sin⁡θ\sin \theta . Basic trigonometric identities include sin⁡2θ+cos⁡2θ=1{\sin ^2}\theta + {\cos ^2}\theta = 1, sec⁡2θ=tan⁡2θ+1{\sec ^2}\theta = {\tan ^2}\theta + 1 and cos⁡ec2θ=cot⁡2θ+1\cos e{c^2}\theta = {\cot ^2}\theta + 1. 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.