Question

How do you simplify $ \tan 35 \times \sec 55 \times \cos 35 $ ?

Answer 1

Hint: In this question, we have to first simplify the given expression by applying the basic identities of trigonometry such as quotient identity of trigonometry which states that $ \tan \theta $ is the ratio of $ \sin \theta $ and $ \cos \theta $ , and can be expressed as $ \dfrac{{\sin \theta }}{{\cos \theta }} $ . After this, we have to apply the reciprocal identity of trigonometry which states that $ \sec \theta $ is the reciprocal of $ \cos \theta $ . And then we will apply the cofunction identity of trigonometry and will write $ \cos \left( {90 - \theta } \right) $ as $ \sin \theta $ to simplify the expression given to us.

Complete step-by-step answer:
(i)We are given to solve the expression:
 $ \tan 35 \times \sec 55 \times \cos 35 $
Since we know through the quotient identity of trigonometry that,
 $ \tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }} $
Therefore, we can write $ \tan 35 $ as $ \dfrac{{\sin 35}}{{\cos 35}} $ . So, our expression will become:
 $ \dfrac{{\sin 35}}{{\cos 35}} \times \sec 55 \times \cos 35 $
(ii)As we can see we have $ \cos 35 $ in the numerator as well as in the denominator, it will get canceled. Therefore, now our expression will become:
 $ \sin 35 \times \sec 55 $
(iii)Now as we know through the reciprocal identity of trigonometry, that:
 $ \sec \theta = \dfrac{1}{{\cos \theta }} $
Therefore, we can write $ \sec 55 $ as $ \dfrac{1}{{\cos 55}} $ . So, our expression will become:
 $ \sin 35 \times \dfrac{1}{{\cos 55}} $
(iv)Now we have $ \cos 55 $ in the denominator which can also be written as:
 $ \cos 55 = \cos \left( {90 - 35} \right) $
As we know through the cofunction identity that:
 $ \cos \left( {90 - \theta } \right) = \sin \theta $
Therefore, if we apply the above stated identity, we can write $ \cos 55 $ as:
 $ \cos 55 = \sin 35 $
Therefore, our expression becomes:
 $ \sin 35 \times \dfrac{1}{{\sin 35}} $
Now since, we have $ \sin 35 $ in the numerator as well as in the denominator, it will be cancelled out and we will get $ 1 $ as the answer.
Hence, the simplification of $ \tan 35 \times \sec 55 \times \cos 35 $ is $ 1 $ .
So, the correct answer is “1”.

Note: We could also first use the reciprocal identity of trigonometry to write $ \sec 55 $ as $ \dfrac{1}{{\cos 55}} $ and then apply the cofunction identity of trigonometry to write the $ \cos 55 $ in the denominator as $ \sin 35 $ . Then our expression would have become $ \tan 35 \times \dfrac{1}{{\sin 35}} \times \cos 35 $ and since we know that $ \dfrac{{\cos \theta }}{{\sin \theta }} $ is $ \cot \theta $ , we would have got $ \tan 35 \times \cot 35 $ and as we know $ \cot \theta $ is the reciprocal of $ \tan \theta $ , we would have ultimately got $ 1 $ as the answer.