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.