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$tan 2 60 ° + 4 cos 2 45 ° + 3 sec 2 30 ° + 5 cos 2 90 ° cosec 30 ° + sec 60 ° − cot 2 30 ° =$ [[1]].

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Detailed Solution

tan 2 60 ° + 4 cos 2 45 ° + 3 sec 2 30 ° + 5 cos 2 90 ° cosec 30 ° + sec 60 ° − cot 2 30 ° = 9

.
Given expression is

tan 2 60 ° + 4 cos 2 45 ° + 3 sec 2 30 ° + 5 cos 2 90 ° cos e c 30 ° + sec 60 ° − cot 2 30 °

.
We know that,

cos 45 ° = 12

sec 30 ° = 23

cos e c 30 ° = 2

tan 60 ° = cot 30 ° = 3

cos 90 ° = 0

sec 60 ° = 2

Substitute the values in the expression and solve,

⇒ 32 + 4122 + 3232 + 502 2 + 2 − 32

Opening squares, we get,

⇒ 3 + 4 × 12 + 3 × 43 + 5 × 0 2 + 2 − 3

⇒ 3 + 24 × 12 + 3 × 43 + 5 × 0 4 − 3

⇒ 3 + 2 + 4 1

⇒ 9

Thus, the value of

tan 2 60 ° + 4 cos 2 45 ° + 3 sec 2 30 ° + 5 cos 2 90 ° cosec 30 ° + sec 60 ° − cot 2 30 ° = 9

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