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Question: If \( A = {30^o} \) , verify that \( \cos 2a = {\cos ^2}A - {\sin ^2}A = 2{\cos ^2}A - 1 \)...

If A=30oA = {30^o} , verify that cos⁡2a=cos⁡2A−sin⁡2A=2cos⁡2A−1\cos 2a = {\cos ^2}A - {\sin ^2}A = 2{\cos ^2}A - 1

Explanation

Solution

Hint : In this question, we need to verify that the expression cos⁡2a=cos⁡2A−sin⁡2A=2cos⁡2A−1\cos 2a = {\cos ^2}A - {\sin ^2}A = 2{\cos ^2}A - 1 is true. For this, we will use the trigonometric identities by substitute the value of A in trigonometric functions.

Complete step by step solution:

Functionsin⁡θ\sin \thetacos⁡θ\cos \thetatan⁡θ\tan \theta
0010
300{30^0}12\dfrac{1}{2}32\dfrac{{\sqrt 3 }}{2}13\dfrac{1}{{\sqrt 3 }}
450{45^0}12\dfrac{1}{{\sqrt 2 }}12\dfrac{1}{{\sqrt 2 }}1
600{60^0}32\dfrac{{\sqrt 3 }}{2}12\dfrac{1}{2}3\sqrt 3
900{90^0}10Indeterminate

In the given function cos⁡2a=cos⁡2A−sin⁡2A=2cos⁡2A−1\cos 2a = {\cos ^2}A - {\sin ^2}A = 2{\cos ^2}A - 1 Substitute the given value of A which is equal 30o{30^o}
We get cos⁡2(30∘)=cos⁡2(30∘)−sin⁡2(30∘)=2cos⁡2(30∘)−1\cos 2\left( {{{30}^ \circ }} \right) = \cos {}^2\left( {{{30}^ \circ }} \right) - {\sin ^2}\left( {{{30}^ \circ }} \right) = 2{\cos ^2}\left( {{{30}^ \circ }} \right) - 1
Now we can write this function as
cos⁡(60∘)=(cos⁡30∘)2−(sin⁡30∘)2=2(cos⁡30∘)2−1\cos \left( {{{60}^ \circ }} \right) = {\left( {\cos {{30}^ \circ }} \right)^2} - {\left( {\sin {{30}^ \circ }} \right)^2} = 2{\left( {\cos {{30}^ \circ }} \right)^2} - 1
Now we will substitute the values of the function for their respective angles from the above given trigonometric table
12=(32)2−(12)2=2(32)2−1\dfrac{1}{2} = {\left( {\dfrac{{\sqrt 3 }}{2}} \right)^2} - {\left( {\dfrac{1}{2}} \right)^2} = 2{\left( {\dfrac{{\sqrt 3 }}{2}} \right)^2} - 1
By further solving this function

12=34−14=2×34−1 12=24=32−1 12=12=3−22 12=12=12   \dfrac{1}{2} = \dfrac{3}{4} - \dfrac{1}{4} = 2 \times \dfrac{3}{4} - 1 \\\ \dfrac{1}{2} = \dfrac{2}{4} = \dfrac{3}{2} - 1 \\\ \dfrac{1}{2} = \dfrac{1}{2} = \dfrac{{3 - 2}}{2} \\\ \dfrac{1}{2} = \dfrac{1}{2} = \dfrac{1}{2} \;

Hence verified, left term, middle term and the right term of the function are equal.

Note : Write the value of the function in the relation for the given respective angles in cases of cosecθ{\text{cosec}}\theta ,sec⁡θ\sec \theta and cot⁡θ\cot \theta , either inverse them or write their values, respectively. The trigonometric function is the function that relates the ratio of the length of two sides with the angles of the right-angled triangle widely used in navigation, oceanography, the theory of periodic functions, and projectiles.